# 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 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 $O$ .

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

$d=2r$ 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 $c$ for the length of this line.

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

 $\pi ={\frac {c}{d}}$ $\therefore {\textrm {(therefore)}}$ $c=2\pi r$ $c=\pi d$  The area of the circle is equal to $\pi$ times the area of the gray square.

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

$A=\pi r^{2}$ ## Calculating π

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

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

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

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

$\pi ={\frac {A}{r^{2}}}$ ## Calculating measures of a circle

### Area

Using the radius: $A=\pi r^{2}={\frac {\tau r^{2}}{2}}$ Using the diameter: $A={\frac {\pi d^{2}}{4}}={\frac {\tau d^{2}}{8}}$ Using the circumference: $A={\frac {c^{2}}{2\tau }}={\frac {c^{2}}{4\pi }}$ ### Circumference

Using the radius: $c=\tau r=2\pi r$ Using the diameter: $c=\pi d={\frac {\tau d}{2}}$ Using the area: $c={\sqrt {2\tau A}}=2{\sqrt {\pi A}}$ ### Diameter

Using the radius: $d=2r$ Using the circumference: $d={\frac {c}{\pi }}={\frac {2c}{\tau }}$ Using the area: $d={\sqrt {\frac {8A}{\tau }}}=2{\sqrt {\frac {A}{\pi }}}$ Using the diameter: $r={\frac {d}{2}}$ Using the circumference: $r={\frac {c}{\tau }}={\frac {c}{2\pi }}$ Using the area: $r={\sqrt {\frac {A}{\pi }}}={\sqrt {\frac {2A}{\tau }}}$ 