# Circle

A Circle
A **circle** is a round, two-dimensional shape. All points on the edge of the circle are at the same distance from the center.

The point on the side. Mathematicians use the letter WIKINIUM\_MATH\_PH\_0 for the length of a circle's radius. The center of a circle is the point in the very middle. It is often written as WIKINIUM\_MATH\_PH\_2 .

The diameter (meaning "all the way across") of a circle is a straight line that goes from one side to the opposite and right through the center of the circle. Mathematicians use the letter WIKINIUM\_MATH\_PH\_4 for the length of this line. The diameter of a circle is equal to twice its radius ( WIKINIUM\_MATH\_PH\_6 equals WIKINIUM\_MATH\_PH\_8 times WIKINIUM\_MATH\_PH\_10 ):

 WIKINIUM\_MATH\_PH\_12
The circumference (meaning "all the way around") of a circle is the line that goes around the center of the circle. Mathematicians use the letter WIKINIUM\_MATH\_PH\_14 for the length of this line.

The number WIKINIUM\_MATH\_PH\_16 (written as the Greek letter *pi*) is a very useful number. It is the length of the circumference divided by the length of the diameter ( WIKINIUM\_MATH\_PH\_18 equals WIKINIUM\_MATH\_PH\_20 divided by WIKINIUM\_MATH\_PH\_22 ). As a fraction the number WIKINIUM\_MATH\_PH\_24 is equal to about WIKINIUM\_MATH\_PH\_26 or WIKINIUM\_MATH\_PH\_28 (which is closer) and as a number it is about WIKINIUM\_MATH\_PH\_30 .

|  |  |
| --- | --- |
|  |  WIKINIUM\_MATH\_PH\_32  |
|  WIKINIUM\_MATH\_PH\_34  |  WIKINIUM\_MATH\_PH\_36  |
|  |  WIKINIUM\_MATH\_PH\_38  |

The area of the circle is equal to WIKINIUM\_MATH\_PH\_40 times the area of the gray square.
The area, WIKINIUM\_MATH\_PH\_42 , inside a circle is equal to the radius multiplied by itself, then multiplied by WIKINIUM\_MATH\_PH\_44 ( WIKINIUM\_MATH\_PH\_46 equals WIKINIUM\_MATH\_PH\_48 times WIKINIUM\_MATH\_PH\_50 times WIKINIUM\_MATH\_PH\_52 ).

 WIKINIUM\_MATH\_PH\_54
## Calculating π

 WIKINIUM\_MATH\_PH\_56 can be measured by drawing a circle, then measuring its diameter ( WIKINIUM\_MATH\_PH\_58 ) and circumference ( WIKINIUM\_MATH\_PH\_60 ). This is because the circumference of a circle is always equal to WIKINIUM\_MATH\_PH\_62 times its diameter.

 WIKINIUM\_MATH\_PH\_64
 WIKINIUM\_MATH\_PH\_66 can also be calculated by only using mathematical methods. Most methods used for calculating the value of WIKINIUM\_MATH\_PH\_68 have desirable mathematical properties. However, they are hard to understand without knowing trigonometry and calculus. However, some methods are quite simple, such as this form of the Gregory-Leibniz series:

 WIKINIUM\_MATH\_PH\_70
While that series is easy to write and calculate, it is not easy to see why it equals WIKINIUM\_MATH\_PH\_72 . A much easier way to approach is to draw an imaginary circle of radius WIKINIUM\_MATH\_PH\_74 centered at the origin. Then any point WIKINIUM\_MATH\_PH\_76 whose distance WIKINIUM\_MATH\_PH\_78 from the origin is less than WIKINIUM\_MATH\_PH\_80 , calculated by the Pythagorean theorem, will be inside the circle:

 WIKINIUM\_MATH\_PH\_82
Finding a set of points inside the circle allows the circle's area WIKINIUM\_MATH\_PH\_84 to be estimated, for example, by using integer coordinates for a big WIKINIUM\_MATH\_PH\_86 . Since the area WIKINIUM\_MATH\_PH\_88 of a circle is WIKINIUM\_MATH\_PH\_90 times the radius squared, WIKINIUM\_MATH\_PH\_92 can be approximated by using the following formula:

 WIKINIUM\_MATH\_PH\_94
## Calculating measures of a circle

### Area

Using the radius: WIKINIUM\_MATH\_PH\_96

Using the diameter: WIKINIUM\_MATH\_PH\_98

Using the circumference: WIKINIUM\_MATH\_PH\_100

### Circumference

Using the radius: WIKINIUM\_MATH\_PH\_102

Using the diameter: WIKINIUM\_MATH\_PH\_104

Using the area: WIKINIUM\_MATH\_PH\_106

### Diameter

Using the radius: WIKINIUM\_MATH\_PH\_108

Using the circumference: WIKINIUM\_MATH\_PH\_110

Using the area: WIKINIUM\_MATH\_PH\_112

### Radius

Using the diameter: WIKINIUM\_MATH\_PH\_114

Using the circumference: WIKINIUM\_MATH\_PH\_116

Using the area: WIKINIUM\_MATH\_PH\_118

## Equations

### Cartesian coordinates

The equation of a circle in Cartesian coordinates, with the centre at WIKINIUM\_MATH\_PH\_120 and radius WIKINIUM\_MATH\_PH\_122 , is given by
 WIKINIUM\_MATH\_PH\_124

### Polar coordinates

A circle centred at the origin with radius WIKINIUM\_MATH\_PH\_126 is the equivalent of a constant function,
 WIKINIUM\_MATH\_PH\_128

A circle with a centre at a different location WIKINIUM\_MATH\_PH\_130 and radius WIKINIUM\_MATH\_PH\_132 can be found by using the conversion between polar and Cartesian coordinates:
 WIKINIUM\_MATH\_PH\_134
Substituting this into the equation for a circle in rectangular coordinates gives
 WIKINIUM\_MATH\_PH\_136
This is a quadratic equation in WIKINIUM\_MATH\_PH\_138 . The quadratic formula can be used:
 WIKINIUM\_MATH\_PH\_140
Whether the square root is added or subtracted does not matter as long as WIKINIUM\_MATH\_PH\_142 is allowed to make a full rotation:
 WIKINIUM\_MATH\_PH\_144
In polar coordinates, the points WIKINIUM\_MATH\_PH\_146 and WIKINIUM\_MATH\_PH\_148 are equal, so this means that both the positive and negative conjugates are included by other half of the domain.

## Related pages

* Concentric
* Semicircle
* Sphere
* Squaring the circle
* Pi
* Tau

## Other websites

* Calculate the measures of a circle online