# 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 ${\displaystyle r}$ 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 ${\displaystyle r}$):[1]

${\displaystyle d=2r}$

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.[2]

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

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

## Calculating π

${\displaystyle \pi }$ 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.[1]

${\displaystyle \pi ={\frac {c}{d}}}$

${\displaystyle \pi }$ 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 ${\displaystyle r}$ centered at the origin. Then any point ${\displaystyle (x,y)}$ whose distance ${\displaystyle d}$ from the origin is less than ${\displaystyle r}$, 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 ${\displaystyle r}$. 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

### 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: ${\displaystyle d=2{\sqrt {\frac {A}{\pi }}}=2{\sqrt {\frac {2A}{\tau }}}}$

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

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

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

## References

1. Weisstein, Eric W. "Circle". mathworld.wolfram.com. Retrieved 2020-09-24.
2. "Basic information about circles (Geometry, Circles)". Mathplanet. Retrieved 2020-09-24.