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

The radius of a circle is a line from the center of the circle to a point on the side. Mathematicians use the letter r 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 {\displaystyle O}.

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 {\displaystyle d} for the length of this line. The diameter of a circle is equal to twice its radius ({\displaystyle d} equals {\displaystyle 2} times r):

{\displaystyle d=2r}
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 {\displaystyle c} for the length of this line.

The number {\displaystyle \pi } (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 ({\displaystyle \pi } equals {\displaystyle c} divided by {\displaystyle d}). As a fraction the number {\displaystyle \pi } is equal to about {\displaystyle {\frac {22}{7}}} or {\displaystyle {\frac {355}{113}}} (which is closer) and as a number it is about {\displaystyle 3.1415926536}.

|  |  |
| --- | --- |
|  | {\displaystyle \pi ={\frac {c}{d}}} |
| {\displaystyle \therefore {\textrm {(therefore)}}} | {\displaystyle c=2\pi r} |
|  | {\displaystyle c=\pi d} |

The area of the circle is equal to {\displaystyle \pi } times the area of the gray square.
The area, {\displaystyle A}, inside a circle is equal to the radius multiplied by itself, then multiplied by {\displaystyle \pi } ({\displaystyle A} equals {\displaystyle \pi } times r times r).

{\displaystyle A=\pi r^{2}}

Calculating π
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{\displaystyle \pi } can be measured by drawing a circle, then measuring its diameter ({\displaystyle d}) and circumference ({\displaystyle c}). This is because the circumference of a circle is always equal to {\displaystyle \pi } times its diameter.

{\displaystyle \pi ={\frac {c}{d}}}
{\displaystyle \pi } can also be calculated by only using mathematical methods. Most methods used for calculating the value of {\displaystyle \pi } 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:

{\displaystyle \pi ={\frac {4}{1}}-{\frac {4}{3}}+{\frac {4}{5}}-{\frac {4}{7}}+{\frac {4}{9}}-{\frac {4}{11}}+\cdots }
While that series is easy to write and calculate, it is not easy to see why it equals {\displaystyle \pi }. A much easier way to approach is to draw an imaginary circle of radius r centered at the origin. Then any point {\displaystyle (x,y)} whose distance {\displaystyle d} from the origin is less than r, calculated by the Pythagorean theorem, will be inside the circle:

{\displaystyle d={\sqrt {x^{2}+y^{2}}}}
Finding a set of points inside the circle allows the circle's area {\displaystyle A} to be estimated, for example, by using integer coordinates for a big r. Since the area {\displaystyle A} of a circle is {\displaystyle \pi } times the radius squared, {\displaystyle \pi } can be approximated by using the following formula:

{\displaystyle \pi ={\frac {A}{r^{2}}}}

Calculating measures of a circle
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### Area

Using the radius: {\displaystyle A=\pi r^{2}={\frac {\tau r^{2}}{2}}}

Using the diameter: {\displaystyle A={\frac {\pi d^{2}}{4}}={\frac {\tau d^{2}}{8}}}

Using the circumference: {\displaystyle A={\frac {c^{2}}{2\tau }}={\frac {c^{2}}{4\pi }}}

### Circumference

Using the radius: {\displaystyle c=\tau r=2\pi r}

Using the diameter: {\displaystyle c=\pi d={\frac {\tau d}{2}}}

Using the area: {\displaystyle c={\sqrt {2\tau A}}=2{\sqrt {\pi A}}}

### Diameter

Using the radius: {\displaystyle d=2r}

Using the circumference: {\displaystyle d={\frac {c}{\pi }}={\frac {2c}{\tau }}}

Using the area: {\displaystyle d=2{\sqrt {\frac {A}{\pi }}}=2{\sqrt {\frac {2A}{\tau }}}}

### Radius

Using the diameter: {\displaystyle r={\frac {d}{2}}}

Using the circumference: {\displaystyle r={\frac {c}{\tau }}={\frac {c}{2\pi }}}

Using the area: {\displaystyle r={\sqrt {\frac {A}{\pi }}}={\sqrt {\frac {2A}{\tau }}}}

Related pages
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* Semicircle
* Sphere
* Squaring the circle
* Pi
* Tau

Other websites
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Wikimedia Commons has media related to ***Circle***.

* Calculate the measures of a circle online

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