Telling Time and Reading Clock Hands | Wyzant Resources Resources / Lessons / Math / Elementary Math / Telling Time Telling Time Being able to tell and work with time is a very valuable skill. In order to tell time, there are several key points to remember. We measure time in seconds, minutes, hours, days, weeks, months, and years. On a clock, we measure time in hours and minutes (and sometimes seconds). On a calendar, we measure time in days, weeks, months, and years. This section will go through telling time on a clock—using minutes and hours. In order to work with minutes and hours, you must remember that there are 60 seconds in a minute, and 60 minutes in an hour. There are 24 hours in one day. Parts of a Clock A clock is a device used to tell time. Clocks measure time in hours and minutes, and have an hour hand (that moves forward one clock number every hour) and a minute hand (that moves forward one tick every minute—it moves forward one clock number every 5 minutes).

A clock is split up into 12 sections, and each section is worth 5 minutes (because 5 x 12 = 60, and there are 60 minutes in an hour). One complete trip around the clock by the minute hand means that one hour has passed. When you read a clock, you look at the hour hand first, and then you look at the minute hand. The hour hand is shorter, and the minute hand is longer—this is how you tell them apart. A normal clock looks like this: The blue numbers are standard on all clocks—these are clock numbers. You will not see the orange numbers on normal clocks the way they are on the outside of this clock, but these are the minute marks—thus, every 1 clock number is equal to 5 minutes. It can be even further broken down like this: Now, you can see that the orange numbers on the outside count from 1 to 60. These numbers represent each minute of an hour. Now, we’ll add in the hour and minute hands. Remember, the hour hand is shorter, and the minute hand is longer. We’re going to set our clock to say 1 o’clock.

Notice that our clock hands have arrows—some clocks have hands with arrows, and others do not. We will show ours with arrows. Reading a Clock Reading a clock takes practice, but after you learn how, it’s very easy to do! First, you’re going to locate the hour hand. The hour hand is the shorter of the two clock hands. It moves from one clock number to the next every hour, and it makes one complete rotation around the clock every 12 hours (since there are 12 clock numbers on each clock). Look at which clock number the hour hand is pointing to. That is the number you will use for the hour. Many times, the arrow will not be pointing exactly at the number; in that case, you would look at which number it was at most recently. For example, if the hour hand is half way between the 2 and the 3, you would use 2 as the hour number. The only way you can use a specific number for the hour number is if it has already passed the number in its rotation around the clock. So, even if the hour hand is very close to the 3, you would still say 2 unless the hand has reached, or passed, the 3.

Here’s an example of what we mean: Here, you can see that the hour hand is a little more than half way between the 1 and the 2. Even though it is closer to the 2, we would still say that 1 is our hour number because the hour hand has not passed 2 yet. Now let’s talk about the minute hand. The minute hand is the longer of the two hands. It points to the minutes that have passed since a new hour began. It makes one rotation around the clock every hour. If the minute hand is pointing to a clock number, that means the number of minutes is a multiple of 5 (5, 10, 15, 20 etc) and you would give that number as the number of minutes. For example, if the minute hand is pointing to the clock number 1, you would say 5 minutes. If the minute hand is pointing to the clock number 6, you would say 30 minutes. You can easily figure this out by counting by 5s around the clock.

However, if the minute hand is in between two clock numbers, you have to count by 5s until you get to the nearest clock number, and then count up by ones to figure out how many minutes you have. Take this as an example: Notice that the hour hand is between the 7 and the 8. To count the minutes, you would count each clock number—remember to count by fives!—until you get to the 7 clock number. You should have counted 35 minutes. Then, you would estimate where the hand is actually pointing. Some clocks have little slashes for each minute, while others do not. In our clock example, we have the numbers written around the edge to make it easier to read. So, in your mind, extend the minute hand all the way to the edge, and read how many more minutes it has passed. In our example, the minute hand is pointing to the 38. Normally, you would see that it’s at the 3rd mark after the clock number, so you would add 3 to 35 and get 38. Since ours is written out, we can just look and see where it’s pointing, and read it that way.

Thus, to tell the time, you would say it is 1:38, which is read “one thirty-eight.” Now, you can practice reading clocks with the following diagram. Once you think you have the answer figured out, type it into the box to see if you’re correct! Oh and don’t forget to include the colon (:) between the hours and minutes, like this: 1:38. The hour hand is between the 6 and the 7, so the hour number is 6. The minute hand is between the 9 and the 10, and it seems that it points closest to 48, so our actual time is 6:48. 6:48 AM and PM Once you can read the hour and minutes, you may also be asked whether it is AM or PM. This is not read from the clock, but rather from what time of day it is. AM runs from midnight (12:00) to 11:59 (one minute before noon). At noon, it becomes PM, and PM goes from noon (12:00) to 11:59 (one minute before midnight).

Many people make generalizations and say that AM is when it’s light outside, and PM is when it’s dark outside. But don’t fall into this trap! That’s not entirely true. At 12:00 AM, it is dark outside, because it is the middle of the night. Around 5:00 or 6:00 AM (depending on the time of year) the sun comes up, and it is light out from about 6:00 AM until about 7:00 or 8:00 PM. Thus, at noon (12 PM), and until around 8:00 PM, it is still light outside! Around 7:00 or 8:00 PM, the sun goes down, and it gets dark out. It gets dark around 8:00 PM, and is dark until the next morning, when the sun comes up around 6:00 AM. As you can see, part of the AM is spent in the dark, and part is spent in the light; part of the PM is spent in the light, and part is spent in the dark!

Sometimes, you’ll be asked to answer questions based on whether the time is AM or PM. For example, what are you doing at 12:00? If it were 12:00 AM, you’d probably be sleeping! However, if it were 12:00 PM, you’d probably be eating lunch! Let’s practice this. Answer the following questions: At 7:00 AM, are you waking up, or getting ready for bed? We know that you would be waking up, because 7 AM would mean it is pretty early in the morning, so you wouldn't be ready to go to bed yet. Waking up At 5:00 PM, are you eating dinner or sleeping? 5:00 PM is late afternoon/early evening, just a few hours before the sun sets. Thus, you would be eating dinner, it wouldn't quite be time for bed yet. {Eating dinner|eating dinner|eating|dinner} At 3:00 PM is it light or dark out? (answer choices: light, dark) 3:00 PM is just a few hours afternoon, and the sun is still shining brightly at this point in the day.

{light|Light} At 2:00 AM is the sun shining? (yes or no) 2:00 AM is just 2 hours after midnight, so it is still considered the middle of the night. The sun doesn't come up until a few hours later, around 6:00 AM. {No|no} Now you’re getting the hang of it! So if someone asks you to tell whether it’s AM or PM when you’re telling time, you’ll be able to. Elapsed Time Elapsed time means how much time has passed from one point to another. Being able to measure elapsed time is important because you will be able to easily figure out starting and ending times of events, as well as how much time you will need, total, to get somewhere, get home, and so on. Measuring elapsed time involves knowing the start time and end time of an event, and then figuring out how much time was in the middle. For example, you might be asked to calculate the elapsed time if you went to the park at 1:00 PM and came back home at 3:00 PM, how much time did you spend at the park?

In order to solve this problem, you would need to figure out how many hours and minutes are between 1:00 and 3:00 PM. The easiest way to do this is to “count up” from 1:00 to 3:00. In order to do this, you would think: from 1:00 PM to 2:00 PM is one hour. From 2:00 PM to 3:00 PM is one hour. Therefore, we add 1 hour + 1 hour = 2 hours total. Thus, 2 hours have elapsed. You can also set up a subtraction problem in order to solve this, it would look like this: As you can see, in order to subtract we put the ending time at the top of our problem, and subtracted our starting time. This is how you would set up any time subtraction problem. Now, let’s try a harder problem. This time, you’re going to a birthday party. The party starts at 1:30 PM and lasts until 4:00 PM. How long does the party last? You have two options in order to solve this problem.

The first option is to count up from 1:30 to 4:00. To do this, you would count by hours first until you get close to 4:00, and then count how many minutes are left. You know that from 1:30 to 2:30 is one hour. From 2:30 to 3:30 is another hour. If you count from 3:30 to 4:30, you will have gone too far, so you would stop counting hours at 3:30. So far, you have 2 hours. Now, you need to count how many minutes you have left. Well, you know that there are 60 minutes in an hour. You’re starting at 30 and going to 60, which means you have 30 minutes to get to 4:00. Now, you add the hours and minutes together, and come up with 2 hours and 30 minutes. The other option in solving this problem is to set up a subtraction problem. It would look like this: Notice that we had to borrow in order to subtract the half hour (30).

When we borrowed from the 4, we crossed out the 4 and made it a 3 (that’s normal) but then instead of putting a 1 in front of the next digit, we added 60 to the minutes of the top number. Now, we can do 60 – 30, and subtract the rest normally, like this: Notice that our answer looks like this: 2:30, but when you report your answer, you would write 2 hr 30 min. The number before the colon is hours, and the number after the colon is minutes. Telling Time Quiz Now let’s try a few practice problems. We’ll give you the start and end time of an event, and you can figure out how much time elapsed during the event. 1. School starts at 8:45 AM. Lunch is at 11:30 AM. How much time elapses between school starting, and eating lunch? (When you answer, make sure you use hrs after the hours number, and min after the minutes) If you used method one, you would count forwards from 8:45 to 11:30.

You would count 8:45 to 9:45 as one hour, then 9:45 to 10:45 as another hour. You would then stop and realize that if you counted an entire hour more, then you would be at 11:45, and would have passed 11:30. Therefore, you know that you need to stop and 10:45 and count forward using minutes for the rest. You know that from 10:45 to 11:00 is 15 minutes, and then from 11:00 to 11:30 is 30 minutes, so you would add those two numbers together, 15 + 30 = 45. Now, go back and remember the 2 hours you had already, so you end with a total of 2 hrs 45 min. If you used method two, you would have written out a subtraction problem starting with 11:30 and subtracting 8:45.

You would have to borrow, and add 60 to the top number, like this: Notice that the 30 of the top number gets 60 added to it, resulting in 90. After that, it’s simply normal subtraction, ending with 2:45. Remember, the number before the colon is hours, the number after the colon is minutes, so your answer is 2 hrs 45 min. 2 hrs 45 min 2. Kim works at in a factory where she has to write down exactly what time she gets to work, and exactly what time she is leaving. On Tuesday, she gets to work at 1:55 PM and leaves work at 9:20 PM. How long was she at work? If you used method one, you would count forwards from 1:55 to 9:20. You would count whole hours from 1:55 to 8:55, which gives you 7 hours. You realize that you cannot go past 8:55 because adding another hour would take you past the time Kim left work. Once you stop at 8:55, you need to count up by the minutes.

From 8:55 to 9:00, there are 5 minutes. From 9:00 to 9:20, there are 20 minutes. When you add these all together, you would get 7 hrs 25 min. The second method would involve setting up a subtraction problem. You would use 9:20 as the top number because that is the ending time. You would use 1:55 as the bottom number, that’s getting subtracted, because that’s the beginning time. You would need to borrow from the 9 and add 60 to the 20 minutes on top. Then your subtraction becomes 80 – 55. You need to borrow again in order to complete this, but this time you would borrow as normal and put a 1 in front of the next digit, like this: Thus, your final answer is 7 hrs 25 min. 7 hrs 25 min Activity: Clocks and Angles About Ads Activity: Clocks and Angles This activity is about Analog clocks and the angles made by the hands of the clock. You can find out more about angles and how they're measured on the page Degrees (Angles) .

What is the angle between the hands of a clock at 1 o'clock? At 1 o'clock the minute hand (red) points to the 12 and the hour hand (blue) points to the 1. So we need to find the angle between the 12 and the 1. How many of this angle are there in a complete turn? There are 12 of them in a complete turn (360°), so each one must be 360° ÷ 12 = 30° So the angle between the hands of a clock at 1 o'clock is 30°. Note: It doesn't matter whether we are talking about 1 am or 1 pm, the answer is exactly the same for both. The angle between the hands at 1 o'clock could also be given as the reflex angle 330°, but we will always give the smaller (acute or obtuse) angle. What is the angle between the hands of a clock at 2:30? At 2:30 the minute hand (red) points to the 6 and the hour hand (blue) points halfway between the 2 and the 3.

So how many lots of 30° do we have this time? The angle between the 5 and the 6 is 30° The angle between the 4 and the 5 is 30° The angle between the 3 and the 4 is 30° The remaining angle is ½ × 30° = 15° So the angle between the hands of a clock at 2:30 = 30° + 30° + 30° + 15° = 105° Complete the following table (give the smaller angle in each case): Time Check your answers at the bottom of the page. More Complicated Times Finding the angle between the hands of a clock is easy as long as we don't use complicated times. For example finding the angle between the hands at 9:37 is much more difficult. You can try that one if you wish, but it's probably too hard. Example: At what times is the angle between the hands of a clock equal to 30°? Notice that the question asks for 'times'. There are many possible answers. Some of them are easy to find, others much more difficult.

Here are two easy answers: But what about this one? At first glance, it looks like this might be a 30° angle also, but at 4:15, the hour hand has already moved a quarter of the way between the 4 and the 5. So the angle is 30° + ¼ × 30° = 30° + 7½° = 37½°. This might be a more accurate answer: Can you find more 30° angles like this one? At what times of the day do the hands of a clock lie in a straight line? In other words the angle between them is 180°? One obvious answer is 6 o'clock: But what other answers could there be? 9:15 is not correct for a similar reason that 4:15 didn't give us exactly a 30° angle ... the hour hand has moved on beyond the 9. This seems to be a very difficult question to answer, but there is an easy way. How many times between 6:00 am and 6:00 pm do the hands make a straight line?

There must be at least one point time for each hour: one between 7:00 am and 8:00 am, one between 8:00 am and 9:00 am, one between 9:00 am and 10:00 am, etc, up to ... one between 4:00 pm and 5:00 pm That makes 11 equal parts, and so: 12 Hours / 11 Word Problem Practice Word Problem Practice You're Reading a Free Preview Pages 5 to 7 are not shown in this preview. You're Reading a Free Preview Pages 12 to 82 are not shown in this preview. This action might not be possible to undo. Are you sure you want to continue? CANCEL We've moved you to where you read on your other device. Get the full title to continue Get the full title to continue reading from where you left off, or restart the preview. Restart preview SparkNotes: GRE: Geometry SparkNotes Geometry Geometry has a long and storied history that goes back thousands of years, which we’re sure you’re dying to hear about.

But unless you’re taking the GRE to go for your masters in math history, this has no possible relevance to your life. So we’ll skip the history lesson and get right to the facts. Lines and Angles An angle is composed of two lines and is a measure of the spread between them. The point where the two lines meet is called the vertex. On the GRE, angles are measured in degrees. Here’s an example: The ° after the 45 is a degree symbol. This angle measures 45 degrees. Acute, Obtuse, and Right Angles Angles that measure less than 90° are called acute angles. Angles that measure more than 90° are called obtuse angles. A special angle that the GRE test makers love is called a right angle. Right angles always measure exactly 90° and are indicated by a little square where the angle measure would normally be, like this: Straight Angles Multiple angles that meet at a single point on a line are called straight angles. The sum of the angles meeting at a single point on a straight line is always equal to 180°.

You may see something like this, asking you to solve for n: The two angles (one marked by 40° and the one marked by n°) meet at the same point on the line. Since the sum of the angles on the line must be 180°, you can plug 40° and n° into a formula like this: 40° + n° = 180° Subtracting 40° from both sides gives n° = 140°, or n = 140. Vertical Angles When two lines intersect, the angles that lie opposite each other, called vertical angles, are always equal. Angles are vertical angles and are therefore equal. Angles and are also vertical, equal angles. Parallel Lines A more complicated version of this figure that you may see on the GRE involves two parallel lines intersected by a third line, called a transversal. Parallel means that the lines run in exactly the same direction and never intersect. Here’s an example: line 1 is parallel to line 2 Don’t assume lines are parallel just because they look like they are—the question will always tell you if two lines are meant to be parallel.

Even though this figure contains many angles, it turns out that it only has two kinds of angles: big angles (obtuse) and little angles (acute). All the big angles are equal to each other, and all the little angles are equal to each other. Furthermore, any big angle + any little angle = 180°. This is true for any figure with two parallel lines intersected by a third line. In the following figure, we label every angle to show you what we mean: As described above, the eight angles created by these two intersections have special relationships to each other: Angles 1, 4, 5, and 8 are equal to one another. Angle 1 is vertical to angle 4, and angle 5 is vertical to angle 8. Angles 2, 3, 6, and 7 are equal to one another. Angle 2 is vertical to angle 3, and angle 6 is vertical to angle 7. The sum of any two adjacent angles, such as 1 and 2 or 7 and 8, equals 180º because they form a straight angle lying on a line.

The sum of any big angle + any little angle = 180°, since the big and little angles in this figure combine into straight lines and all the big angles are equal and all the little angles are equal. So a big and little angle don’t need to be next to each other to add to 180°; any big plus any little will add to 180°. For example, since angles 1 and 2 sum to 180º, and since angles 2 and 7 are equal, the sum of angles 1 and 7 also equals 180º. By using these rules, you can figure out the degrees of angles that may seem unrelated. For example: If line 1 is parallel to line 2, what is the value of y in the figure above? Again, this figure has only two kinds of angles: big angles (obtuse) and little angles (acute). We know that 55° is a little angle, so y must be a big angle.

Since big angle + little angle = 180°, you can write: y° + 55° = 180° y = 125 Perpendicular Lines Two lines that meet at a right angle are called perpendicular lines. If you’re told two lines are perpendicular, just think 90°. We’d love to tell you more about these beauties, but there’s really not much else to say. Polygon Basics A polygon is a two-dimensional figure with three or more straight sides. Polygons are named according to the number of sides they have. Number of Sides n-gon All polygons, no matter how many sides they possess, share certain characteristics: The sum of the interior angles of a polygon with n sides is (n – 2)180°. For instance, the sum of the interior angles of an octagon is (8 – 2)180°= 6(180°) = 1080°. A polygon with equal sides and equal interior angles is a regular polygon. The sum of the exterior angles of any polygon is 360°. The perimeter of a polygon is the sum of the lengths of its sides.

The area of a polygon is the measure of the area of the region enclosed by the polygon. Each polygon tested on the GRE has its own unique area formula, which we’ll cover below. For the most part, the polygons tested in GRE math include triangles and quadrilaterals. All triangles have three sides, but there are special types of triangles that we’ll cover next. Then we’ll move on to four common quadrilaterals (four-sided figures) that appear on the test: rectangles, squares, parallelograms, and trapezoids. We’ll then leave the world of polygons and make our way to circles and then conclude this geometry section with a discussion of the three-dimensional solids you may see on your test: rectangular solids, cubes, and right circular cylinders. Triangles Of all the geometric shapes, triangles are among the most commonly tested on the GRE. But since the GRE tends to test the same triangles over and over, you just need to master a few rules and a few diagrams. We’ll look at some special triangles shortly, but first we’ll explain four very special rules. The Four Rules of Triangles Commit these four rules to memory. 1. The Rule of Interior Angles.

The sum of the interior angles of a triangle always equals 180°. Interior angles are those on the inside. Whenever you’re given two angles of a triangle, you can use this formula to calculate the third angle. For example: What is the value of a in the figure above? Since the sum of the angles of a triangle equals 180°, you can set up an equation: a° + 100° + 60° = 180° Isolating the variable and solving for a gives a = 20. 2. The Rule of Exterior Angles. An exterior angle of a triangle is the angle formed by extending one of the sides of the triangle past a vertex (or the intersection of two sides of a figure). In the figure below, d is the exterior angle. Since, together, d and c form a straight angle, they add up to 180°: d + c = 180°. According to the first rule of triangles, the three angles of a triangle always add up to 180°, so a + b + c = 180°.

Since d + c = 180° and a + b + c = 180°, d must be equal to a + b (the remote interior angles). This generalizes to all triangles as the following rules: The exterior angle of a triangle plus the interior angle with which it shares a vertex is always 180°. The exterior angle is also equal to the sum of the measures of the remote interior angles. 3. The Rule of the Sides. The length of any side of a triangle must be greater than the difference and less than the sum of the other two sides. In other words: difference of other two sides < one side < sum of other two sides Although this rule won’t allow you to determine a precise length of the missing side, it will allow you to determine a range of values for the missing side, which is exactly what the test makers would ask for in such a problem. Here’s an example: What is the range of values for x in the triangle above?

Since the difference of 10 and 4 is 6, and the sum of 10 and 4 is 14, we can determine the range of values of x: 6 < x < 14 Keep in mind that x must be inside this range. That is, x could not be 6 or 14. 4. The Rule of Proportion. In every triangle, the longest side is opposite the largest angle and the shortest side is opposite the smallest angle. Take a look at the following figure and try to guess which angle is largest. In this figure, side a is clearly the longest side and is the largest angle. Meanwhile, side c is the shortest side and is the smallest angle. So c < b < a and This proportionality of side lengths and angles holds true for all triangles. Use this rule to solve the question below: What is one possible value of x if ? (E) 15 According to the rule of proportion, the longest side of a triangle is opposite the largest angle, and the shortest side of a triangle is opposite the smallest angle. The question tells us that angle C < angle A < angle B.

So, the largest angle in triangle ABC is angle B, which is opposite the side of length 8. We know too that the smallest angle is angle C, since angle C < angle A. This means that the third side, with a length of x, measures between 6 and 8 units in length. The only choice that fits this criterion is 7, choice C. Isosceles Triangles An isosceles triangle has two equal sides and two equal angles, like this: The tick marks indicate that sides a and b are equal, and the curved lines inside the triangle indicate that angle A equals angle B. Notice that the two equal angles are the ones opposite the two equal sides. Let’s see how we might put this knowledge to use on the test. Check out this next triangle: With two equal sides, this is an isosceles triangle. Even though you’re not explicitly told that it has two equal angles, any triangle with two equal sides must also have two equal angles. This means that x must be 70°, because angles opposite equal sides are equal.

Knowing that the sum of the angles of any triangle is 180°, we can also calculate y: y + 70 + 70 = 180, or y = 40. Equilateral Triangles An equilateral triangle has three equal sides and three equal angles, like this: The tick marks tip us off that the three sides are equal. We can precisely calculate the angles because the 180° of an equilateral triangle broken into three equal angles yields 60° for each. As soon as you’re given one side of an equilateral triangle, you’ll immediately know the other two sides, because all three have the same measure. If you ever see a triangle with three equal sides, you’ll immediately know its angles measure 60°. Conversely, if you see that a triangle’s three interior angles all measure 60°, then you’ll know its sides must all be equal. Right Triangles A right triangle is any triangle that contains a right angle. The side opposite the right angle is called the hypotenuse. The other two sides are called legs. The angles opposite the legs of a right triangle add up to 90°.

That makes sense, because the right angle itself is 90° and every triangle contains 180° total, so the other two angles must combine for the other 90°. Right triangles are so special that they even have their own rule, known as the Pythagorean theorem. The Pythagorean Theorem The ancient Greeks spent a lot of time philosophizing, eating grapes, and riding around on donkeys. They also enjoyed the occasional mathematical epiphany. One day, Pythagoras discovered that the sum of the squares of the two legs of a right triangle is equal to the square of the hypotenuse. “Eureka!” he shouted, and the GRE had a new topic to test. And the Greek police had a new person to arrest, since Pythagoras had this epiphany while sitting in the bath and immediately jumped out to run down the street and proclaim it to the world, entirely in the buff. Or maybe that was Archimedes . . . ? Hey, it was a long time ago. Suffice it to say that some naked Greek guy thought up something cool. Since Pythagoras took the trouble to invent his theorem, the least we can do is learn it.

Besides, it’s one of the most famous theorems in all of math, and it is tested with regularity on the GRE, to boot. Here it is: In a right triangle, a2 + b2 = c2, where c is the length of the hypotenuse and a and b are the lengths of the two legs. And here’s a simple application: What is the value of b in the triangle above? The little square in the lower left corner lets you know that this is a right triangle, so you’re clear to use the Pythagorean theorem. Substituting the known lengths into the formula gives: 32 + b2 = 52 b2 = 16 b = 4 This is therefore the world-famous 3-4-5 right triangle. Thanks to the Pythagorean theorem, if you know the measures of any two sides of a right triangle, you can always find the third. “Eureka!” indeed. Pythagorean Triples. Because right triangles obey the Pythagorean theorem, only a specific few have side lengths that are all integers. For example, a right triangle with legs of length 3 and 5 has a hypotenuse of length .

Positive integers that obey the Pythagorean theorem are called Pythagorean triples, and these are the ones you’re likely to see on your test as the lengths of the sides of right triangles. Here are some common ones: {3, 4, 5} {7, 24, 25} {8, 15, 17} In addition to these Pythagorean triples, you should also watch out for their multiples. For example, {6, 8, 10} is a Pythagorean triple, since it is a multiple of {3, 4, 5}, derived from simply doubling each value. This knowledge can significantly shorten your work in a problem like this: What is the value of z in the triangle above? Sure, you could calculate it out using the Pythagorean theorem, but who wants to square 120 and 130 and then work the results into the formula? Pythagoras himself would probably say, “Ah, screw it . . .” and head off for a chat with Socrates.

(Actually, Pythagoras died about twenty years before Socrates was born, but you get the point.) But armed with our Pythagorean triples, it’s no problem for us. The hypotenuse is 130, and one of the legs is 120. The ratio between these sides is 130:120, or 13:12. This exactly matches the {5, 12, 13} Pythagorean triple. So we’re missing the 5 part of the triple for the other leg. However, since the sides of the triangle in the question are 10 times longer than those in the {5, 12, 13} triple, the missing side must be 5 × 10 = 50. 30-60-90 Right Triangles As you can see, right triangles are pretty darn special. But there are two extra-special ones that appear with astounding frequency on the GRE. They are 30-60-90 right triangles and 45-45-90 right triangles.

When you see one of these, instead of working out the Pythagorean theorem, you’ll be able to apply standard ratios that exist between the length of the sides of these triangles. The guy who named 30-60-90 triangles didn’t have much of an imagination— or maybe he just didn’t have a cool name like “Pythagoras.” The name derives from the fact that these triangles have angles of 30°, 60°, and 90°. So, what’s so special about that? This: The side lengths of 30-60-90 triangles always follow a specific pattern. If the short leg opposite the 30° angle has length x, then the hypotenuse has length 2x, and the long leg, opposite the 60° angle, has length . Therefore: The sides of every 30-60-90 triangle will follow the ratio 1: :2 Thanks to this constant ratio, if you know the length of just one side of the triangle, you’ll immediately be able to calculate the lengths of the other two.

If, for example, you know that the side opposite the 30º angle is 2 meters, then by using the ratio you can determine that the hypotenuse is 4 meters, and the leg opposite the 60º angle is meters. 45-45-90 Right Triangles A 45-45-90 right triangle is a triangle with two angles of 45° and one right angle. It’s sometimes called an isosceles right triangle, since it’s both isosceles and right. Like the 30-60-90 triangle, the lengths of the sides of a 45-45-90 triangle also follow a specific pattern. If the legs are of length x (the legs will always be equal), then the hypotenuse has length . The sides of every 45-45-90 triangle will follow the ration of This ratio will help you when faced with triangles like this: This right triangle has two equal sides, which means the two angles other than the right angle must be 45° each. So we have a 45-45-90 right triangle, which means we can employ the ratio.

But instead of being 1 and 1, the lengths of the legs are 5 and 5. Since the lengths of the sides in the triangle above are five times the lengths in the right triangle, the hypotenuse must be , or The formula for the area of a triangle is: area = or Keep in mind that the base and height of a triangle are not just any two sides of a triangle. The base and height must be perpendicular, which means they must meet at a right angle. Let’s try an example. What’s the area of this triangle? Note that the area is not , because those two sides do not meet at a right angle. To calculate the area, you must first determine b. You’ll probably notice that this is the 3-4-5 right triangle you saw earlier, so b = 4. Now you have two perpendicular sides, so you can correctly calculate the area as follows: Triangles are surely important, but they aren’t the only geometric figures you’ll come across on the GRE. We move now to quadrilaterals, which are four-sided figures. The first two we cover you’re no doubt familiar with, no matter how long you’ve been out of high school.

The other two you may have forgotten about long ago. Rectangles A rectangle is a quadrilateral in which the opposite sides are parallel and the interior angles are all right angles. The opposite sides of a rectangle are equal, as indicated in the figure below: Area of a Rectangle The formula for the area of a rectangle is: area = base × height or simply a = bh Since the base is the length of the rectangle and the height is the width, just multiply the length by the width to get the area of a rectangle. Diagonals of a Rectangle The two diagonals of a rectangle are always equal to each other, and either diagonal through the rectangle cuts the rectangle into two equal right triangles. In the figure below, the diagonal BD cuts rectangle ABCD into congruent right triangles BAD and BCD. Congruent means that those triangles are exactly identical. Since the diagonal of the rectangle forms right triangles that include the diagonal and two sides of the rectangle, if you know two of these values, you can always calculate the third with the Pythagorean theorem. For example, if you know the side lengths of the rectangle, you can calculate the length of the diagonal. If you know the diagonal and one side length, you can calculate the other side length.

Also, keep in mind that the diagonal might cut the rectangle into a 30-60-90 triangle, in which case you could use the 1: : 2 ratio to make your calculating job even easier. Squares “Hey, buddy, don’t you be no square . . .” Elvis Presley, Jailhouse Rock We don’t know why, but sometime around the 1950s the word square became synonymous with “uncool.” Of course, that’s the same decade that brought us words like daddy-O and dances called “The Hand Jive.” We think that squares are in fact nifty little geometric creatures. They, along with circles, are perhaps the most symmetrical shapes in the universe—nothing to sneeze at. A square is so symmetrical because its angles are all 90º and all four of its sides are equal in length. Rectangles are fairly common, but a perfect square is something to behold. Like a rectangle, a square’s opposite sides are parallel and it contains four right angles. But squares one-up rectangles by virtue of their equal sides.

As if you’ve never seen one of these in your life, here’s what a square looks like: Area of a Square The formula for the area of a square is: area = s2 In this formula, s is the length of a side. Since the sides of a square are all equal, all you need is one side to figure out a square’s area. Diagonals of a Square The square has two more special qualities: Diagonals bisect each other at right angles and are equal in length. Diagonals bisect the vertex angles to create 45º angles. (This means that one diagonal will cut the square into two 45-45-90 triangles, while two diagonals break the square into four 45-45-90 triangles.) Because a diagonal drawn into the square forms two congruent 45-45-90 triangles, if you know the length of one side of the square, you can always calculate the length of the diagonal: Since d is the hypotenuse of the 45-45-90 triangle that has legs of length 5, according to the ratio , you know that .

Similarly, if you know only the length of the diagonal, you can use the same ratio to work backward to calculate the length of the sides. Parallelograms A parallelogram is a quadrilateral whose opposite sides are parallel. That means that rectangles and squares qualify as parallelograms, but so do four-sided figures that don’t contain right angles. In a parallelogram, opposite sides are equal in length. That means that in the figure above, BC = AD and AB = DC. Opposite angles are equal: ∠ABC = ∠ADC and ∠BAD = ∠BCD. Adjacent angles are supplementary, which means they add up to 180°. Here, an example is ∠ABC + ∠BCD = 180°. Area of a Parallelogram The area of a parallelogram is given by the formula: area = bh In this formula, b is the length of the base, and h is the height. As shown in the figure below, the height of a parallelogram is represented by a perpendicular line dropped from one side of the figure to the side designated as the base.

Diagonals of a Parallelogram The diagonals of a parallelogram bisect (split) each other: BE = ED and AE = EC One diagonal splits a parallelogram into two congruent triangles: ∆ABD = ∆BCD Two diagonals split a parallelogram into two pairs of congruent triangles: ∆AEB = ∆DEC and ∆BEC = ∆AED Trapezoids A trapezoid may sound like a new Star Wars character, but it’s actually the name of a quadrilateral with one pair of parallel sides and one pair of nonparallel sides. Here’s an example: In this trapezoid, AB is parallel to CD (shown by the arrow marks), whereas AC and BD are not parallel. Area of a Trapezoid The formula for the area of a trapezoid is a bit more complex than the area formulas you’ve seen for the other quadrilaterals in this section. We’ll set it off so you can take a good look at it: In this formula, s1 and s2 are the lengths of the parallel sides (also called the bases of the trapezoid), and h is the height.

In a trapezoid, the height is the perpendicular distance from one base to the other. If you come across a trapezoid question on the GRE, you may need to use your knowledge of triangles to solve it. Here’s an example of what we mean: Find the area of the figure above. First of all, we’re not told that the figure is a trapezoid, but we can infer as much from the information given. Since both the line labeled 6 and the line labeled 10 form right angles with the line connecting them, the 6 and 10 lines must be parallel. Meanwhile, the other two lines (the left and right sides of the figure) cannot be parallel because one connects to the bottom line at a right angle, while the other connects with that line at a 45° angle. So we can deduce that the figure is a trapezoid, which means the trapezoid area formula is in play. The bases of the trapezoid are the parallel sides, and we’re told their lengths are 6 and 10.

So far so good, but to find the area, we also need to find the trapezoid’s height, which isn’t given. To do that, split the trapezoid into a rectangle and a 45-45-90 triangle by drawing in the height. Once you’ve drawn in the height, you can split the base that’s equal to 10 into two parts: The base of the rectangle is 6, and the leg of the triangle is 4. Since the triangle is 45-45-90, the two legs must be equal. This leg, though, is also the height of the trapezoid. So the height of the trapezoid is 4. Now you can plug the numbers into the formula: Another way to find the area of the trapezoid is to find the areas of the triangle and the rectangle, then add them together: area = 8 + 24 = 32 Circles A circle is the collection of points equidistant from a given point, called the center. Circles are named after their center points—that’s just easier than giving them names like Ralph and Betty. All circles contain 360°.

The distance from the center to any point on the circle is called the radius (r). Radius is the most important measurement in a circle, because if you know a circle’s radius, you can figure out all its other characteristics, such as its area, diameter, and circumference. We’ll cover all that in the next few pages. The diameter (d) of a circle stretches between endpoints on the circle, passing through the center. A chord also extends from endpoint to endpoint on the circle, but it does not necessarily pass through the center. In the following figure, point C is the center of the circle, r is the radius, and AB is a chord. Tangent Lines Tangents are lines that intersect a circle at only one point. Just like everything else in geometry, tangent lines are defined by certain fixed rules. Here’s the first: A radius whose endpoint is the intersection point of the tangent line and the circle is always perpendicular to the tangent line, as shown in the following figure: And the second rule: Every point in space outside the circle can extend exactly two tangent lines to the circle. The distances from the origin of the two tangents to the points of tangency are always equal. In the figure below, XY = XZ.

Central Angles and Inscribed Angles An angle whose vertex is the center of the circle is called a central angle. The degree of the circle (the slice of pie) cut by a central angle is equal to the measure of the angle. If a central angle is 25º, then it cuts a 25º arc in the circle. An inscribed angle is an angle formed by two chords originating from a single point. An inscribed angle will always cut out an arc in the circle that is twice the size of the degree of the inscribed angle. For example, if an inscribed angle is 40º, it will cut an arc of 80º in the circle. If an inscribed angle and a central angle cut out the same arc in a circle, the central angle will be twice as large as the inscribed angle. Circumference of a Circle The circumference is the perimeter of the circle—that is, the total distance around the circle. The formula for circumference of a circle is: circumference = 2πr In this formula, r is the radius. Since a circle’s diameter is always twice its radius, the formula can also be written c = πd, where d is the diameter.

Let’s find the circumference of the circle below: Plugging the radius into the formula, c = 2πr = 2π (3) = 6π. Arc Length An arc is a part of a circle’s circumference. An arc contains two endpoints and all the points on the circle between the endpoints. By picking any two points on a circle, two arcs are created: a major arc, which is by definition the longer arc, and a minor arc, the shorter one. Since the degree of an arc is defined by the central or inscribed angle that intercepts the arc’s endpoints, you can calculate the arc length as long as you know the circle’s radius and the measure of either the central or inscribed angle. The arc length formula is: arc length = In this formula, n is the measure of the degree of the arc, and r is the radius. This makes sense, if you think about it: There are 360° in a circle, so the degree of an arc divided by 360 gives us the fraction of the total circumference that arc represents. Multiplying that by the total circumference (2pr) gives us the length of the arc.

Here’s the sort of arc length question you might see on the test: Circle D has radius 9. What is the length of arc AB? To figure out the length of arc AB, we need to know the radius of the circle and the measure of ∠C, the inscribed angle that intercepts the endpoints of arc AB. The question provides the radius of the circle, 9, but it throws us a little curveball by not providing the measure of ∠C. Instead, the question puts ∠C in a triangle and tells us the measures of the other two angles in the triangle. Like we said, only a little curveball: You can easily figure out the measure of ∠C because, as you know by now, the three angles of a triangle add up to 180º: ∠c = 180° – (50° + 70°) ∠c = 180° – 120° ∠c = 60° Since ∠c is an inscribed angle, arc AB must be twice its measure, or 120º.

Now we can plug these values into the formula for arc length: Area of a Circle If you know the radius of a circle, you can figure out its area. The formula for area is: area = πr2 In this formula, r is the radius. So when you need to find the area of a circle, your real goal is to figure out the radius. In easier questions the radius will be given. In harder questions, they’ll give you the diameter or circumference and you’ll have to use the formulas for those to calculate the radius, which you’ll then plug into the area formula. Area of a Sector A sector of a circle is the area enclosed by a central angle and the circle itself. It’s shaped like a slice of pizza. The shaded region in the figure on the left below is a sector. The figure on the right is a slice of pepperoni pizza. See the resemblance? The area of a sector is related to the area of a circle just as the length of an arc is related to the circumference. To find the area of a sector, find what fraction of 360° the sector makes up and multiply this fraction by the total area of the circle.

In formula form: area of a sector = In this formula, n is the measure of the central angle that forms the boundary of the sector, and r is the radius. An example will help. Find the area of the sector in the figure below: The sector is bounded by a 70° central angle in a circle whose radius is 6. Using the formula, the area of the sector is: Mish-Mashes: Figures with Multiple Shapes The GRE test makers evidently feel it would be no fun if all of these geometric shapes you’re learning appeared in isolation, so sometimes they mash them together. The trick in these problems is to understand and be able to manipulate the rules of each figure individually, while also recognizing which elements of the mish-mashes overlap. For example, the diameter of a circle may also be the side of a square, so if you use the rules of circles to calculate that length, you can then use that answer to determine something about the square, such as its area or perimeter. Mish-mash problems often combine circles with other figures. Here’s an example: What is the length of minor arc BE in circle A if the area of rectangle ABCD is 18?

To find the length of minor arc BE, you have to know two things: the radius of the circle and the measure of the central angle that intersects the circle at points B and E. Because ABCD is a rectangle, and rectangles only have right angles, ∠BAD is 90°. In this question, they tell you as much by including the right-angle sign. But in a harder question, they’d leave the right-angle sign out and expect you to deduce that ∠BAD is 90° on your own. And since that angle also happens to be the central angle of circle A intercepting the arc in question, we can determine that arc BE measures 90°. Finding the radius requires a bit of creative visualization as well, but it’s not so hard. The key is to realize that the radius of the circle is equal to the width of the rectangle. So let’s work backward from the rectangle to give us what we need to know about the circle. The area of the rectangle is 18, and its length is 6. Since the area of a rectangle is simply its length multiplied by its width, we can divide 18 by 6 to get a width of 3.

As we’ve seen, this rectangle width doubles as the circle’s radius, so we’re in business: radius = 3. All we have to do is plug in the values we found into the arc length formula, and we’re done. That covers the two-dimensional figures you should know, but there are also some three-dimensional figures you may be asked about as well. We’ll finish up this geometry section with a look at those. Rectangular Solids A rectangular solid is a prism with a rectangular base and edges that are perpendicular to its base. In English, it looks a lot like a cardboard box. A rectangular solid has three important dimensions: length (l), width (w), and height (h). If you know these three measurements, you can find the solid’s volume, surface area, and diagonal length. Volume of a Rectangular Solid The formula for the volume of a rectangular solid builds on the formula for the area of a rectangle. As discussed earlier, the area of a rectangle is equal to its length times its width. The formula for the volume of a rectangular solid adds the third dimension, height, to get: volume = lwh Here’s a good old-fashioned example: What is the volume of the figure presented below?

The length is 3x, the width is x, and the height is 2x. Just plug the values into the volume formula and you’re good to go: v = (3x)(x)(2x) = 6x3. Surface Area of a Rectangular Solid The surface area of a solid is the area of its outermost skin. In the case of rectangular solids, imagine a cardboard box all closed up. The surface of that closed box is made of six rectangles: The sum of the areas of the six rectangles is the surface area of the box. To make things even easier, the six rectangles come in three congruent pairs. We’ve marked the congruent pairs by shades of gray in the image below: One pair is clear, one pair is light gray, and one pair is dark gray. Two faces have areas of l × w, two faces have areas of l × h, and two faces have areas of w × h. The surface area of the entire solid is the sum of the areas of the congruent pairs: surface area = 2lw + 2lh + 2wh. Let’s try the formula out on the same solid we saw above.

Find the surface area of this: Again, the length is 3x, the width is x, and the height is 2x. Plugging into the formula, we get: surface area = 2(3x)(x) + 2(3x)(2x) + 2(x)(2x) =6x2 + 12x2 + 4x2 = 22x2 Dividing Rectangular Solids If you’re doing really well on the Math section, the CAT program will begin scrounging for the toughest, most esoteric problems it can find to throw your way. One such problem may describe a solid, give you all of its measurements, and then tell you that the box has been cut in half, like so: A number of possible questions could be created from this scenario. For example, you may be asked to find the combined surface area of the two new boxes. Or maybe a Quantitative Comparison question would ask you to compare the volume of the original solid with that of the two new ones. Actually, the volume remains unchanged, but the surface area increases because two new sides (shaded in the diagram) emerge when the box is cut in half.

You may need to employ a bit of reasoning along with the formulas you’re learning to answer a difficult question like this, but it helps to know this general rule: Whenever a solid is cut into smaller pieces, its surface area increases, but its volume is unchanged. Diagonal Length of a Rectangular Solid The diagonal of a rectangular solid, d, is the line segment whose endpoints are opposite corners of the solid. Every rectangular solid has four diagonals, each with the same length, that connect each pair of opposite vertices. Here’s one diagonal drawn in: It’s possible that a question will test to see if you can find the length of a diagonal. Here’s the formula: Again, l is the length, w is the width, and h is the height. The formula is like a pumped-up version of the Pythagorean theorem. Check it out in action: What is the length of diagonal AH in the rectangular solid below if AC = 5, GH = 6, and CG = 3?

The question gives the length, width, and height of the rectangular solid, so you can just plug those numbers into the formula: The problem could be made more difficult if it forced you to first calculate some of the dimensions before plugging them into the formula. Cubes A cube is a three-dimensional square. The length, width, and height of a cube are equal, and each of its six faces is a square. Here’s what it looks like—pretty basic: Volume of a Cube The formula for finding the volume of a cube is essentially the same as the formula for the volume of a rectangular solid: We just need to multiply the length, width, and height. However, since a cube’s length, width, and height are all equal, the formula for the volume of a cube is even easier: volume = s3 In this formula, s is the length of one edge of the cube. Surface Area of a Cube Since a cube is just a rectangular solid whose sides are all equal, the formula for finding the surface area of a cube is the same as the formula for finding the surface area of a rectangular solid, except with s substituted in for l, w, and h.

This boils down to: volume = 6s2 Diagonal Length of a Cube The formula for the diagonal of a cube is also adapted from the formula for the diagonal length of a rectangular solid, with s substituted for l, w, and h. This yields , which simplifies to: volume = Right Circular Cylinders A right circular cylinder looks like one of those cardboard things that toilet paper comes on, except it isn’t hollow. It has two congruent circular bases and looks like this: The height, h, is the length of the line segment whose endpoints are the centers of the circular bases. The radius, r, is the radius of its base. For the GRE, all you need to know about a right circular cylinder is how to calculate its volume. Volume of a Right Circular Cylinder The volume of this kind of solid is the product of the area of its base and its height. Because a right circular cylinder has a circular base, its volume is equal to the area of the circular base times the height or: volume = πr2h Find the volume of the cylinder below: This cylinder has a radius of 4 and a height of 6.

Using the volume formula, its volume = π(4)2(6) = 96π. We’ve covered a lot of ground so far in this Math 101 chapter, working our way through arithmetic, algebra, and now geometry. We’ll bring it on home with our final subject, data analysis. Dice Dice ... ... The dice have 21 points. The numbers 1 to 6 appear on the six sides of a cube. - You throw the dice by hand or by a leather dice cup. You turn the cup upside down and you let the dice roll to a standstill. It is by chance which number appears on top. This is the attraction of throwing the dice. The dice are used in many board games. They often determine the playing. They give the game unexpected turns and add excitement to the games. There are 30 Dice top ... ... If you give the numbers 1, 2, 3, 4, 5, and 6 and form all permutations of the six numbers, you get 1*2*3*4*5*6=6!=720 cubes.

Now there are also those dice among them, which are the same, because you can get them by turning around 13 axes. There are 24 turnings. Hence you only have 720:24 =30 different cubes. The following drawing illustrates this. (The numbers under the cubes give you the number of turnings.) Appearance of the Standard Die top The die has six sides for six numbers. If you try all possible ways to spread them out, you get 30 dice. - Every die has its reflection. You can find the possibilities if you reverse the numbers beside the 1. ... ... You can mark every die by a number with six digits: Start with 1, go upwards, go around 1 counter clockwise and add the number that appears opposite the number 1 on the die. - The red die has 123546 for instance. The dice on the left are ordered by these numbers. You find more about the number 30 at my web page Macmahons Coloured Cubes . There is only the red die in the Western World. ... This die has two characteristics: >The sides opposite each other always add to 7.

>If you look at the corner with the numbers 1, 2, and 3 they are arranged anticounter clockwise. Most of the dice makers follow these rules. ... ...... There is a way the dice can be different. There are two ways of placing the points of number 2, 3, or 6, which change into themselves by turning through 180°. This leads to eight pictures. The red die is the most frequent. But I also found the green ones. Four puzzles with dice will follow now. Guessing the Sum top ... Build a tower of several dice. Is it possible to find the sum of the points of the hidden horizontal cube sides? In this case the tower is formed by three dice. Five surfaces would be counted: The under side of the top die, and the top and bottom sides of the two lower dice. Solution: You take 1 at the top and count the number of the lower cubes. Number 6 is opposite the 1 side. You add 7 for each die. The sum is 6+2x7=20. ... You can make a nice cube puzzle, if you cut the die in nine bars.

The vertical bars are in the middle layer, the horizontal ones on the outside. Then the puzzle is more interesting. ... ... You can also form a 3x3x3-cube with 9 V-Triominos (drawing) or with Soma cubes. Draw points on them. Rubik's Dice top Rubik's dice is a black plastic dice 7cm x 7cm x 7cm with circular holes instead of points. There are seven white squared tiles with red circles, which you can lay inside the cube. The tiles directly lying at the holes stick to walls inside. You can remove them with two connected sticks. The tiles close the holes in red or in white. You have found a solution, when all points have become white. You will soon notice, that you can`t solve the puzzle by trying. There are too many possibilities with seven tiles for closing the holes. Moving the tiles is difficult, too. Act like this: You can look inside the cube and can recognize the patterns of both sides of the tiles. Draw them. The tiles show a number with two digits on the top right (meaning?), which can be used as names. There is the following image: Now the solution is simple.

If you like, don't read further and solve the puzzle yourself... Solution: Tile 78 (left side) only fits on the six, then tile 65 (left side) only fits on the five, then tile 34 (left side) only fits on the four, then tile 58 (right side) only fits on the three, then tile 12 (right side) only fits on the two, and then tile 14 (left side) only fits on the one. Tile 47 is useless. There is only one solution. In the meantime I opened the cube with a saw :-( to scan one side of a tile. Besides I was curious to see how the tiles stick inside. The tiles are made of iron. They are held by six ring magnets. They are inside the cube in the centres. You can see their contours from outside. ... ... Give eight dice. Build a 2x2x2 cube, so that the sum of the points on each side is the same. Here is one of 20 736 solutions with the sum 14. You find more at the German magazine <bild der Wissenschaft> 3-1980.

More Dice Shapes top The cube belongs to the five Platonic solids. These are solids, which are only formed by regular polygons. They are called tetrahedron, hexahedron (cube), octahedron, dodecahedron, and icosahedron. You can build dice after their model. You can find 4,6, 8,12 or 20 numbers by throwing the dice. The number at the top counts. The tetrahedron is an exception. The hidden number counts or the one, that is on top as seen in the scan. I found more interesting solids, which are possible to play the dice from the mathematical view. They are convex, are formed by congruent polygons and have the same position related to the centre of gravity. ("Properties of Dice", URL below). There are more dice forms. I chose two of them. ... ... The die d10 is interesting. It has the shape of a double pyramid with a pentagon as the mutual base. It shows the numbers 0 to 9, the digits of the decimal system. So you can find random numbers with several digits by several dice of this kind. ... ....

The die in ball form is funny: There is an iron ball inside the die, which only rest in one of the six hollows. So the points lie at the top. ... ... There are many possibilities to find random numbers. You can lay six balls marked with number 1 to 6 in a box similar to the national lottery. You can mix them and then take one without looking at. The chosen number is the random number. If you play the "dice" a second time, you must put back the ball in the box first. This is the normal model for a die in mathematics to classify the die in a theory. In spite of the inflation of dice forms: The normal d6 is and will stay the standard die. It has several advantages: >The number 6 is not too small and not too large. You can play well with six random numbers. >The normal die rolls best, especially if the corners are rounded. It is common. >You can make them easily, because the cube sides are perpendicular to each other.

Question: On a regular clockface which number lies opposite 2?
Answer: 8