Circle: Simple curve of Euclidean geometry

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 ${\displaystyle r}$):

${\displaystyle d=2r}$

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, ${\displaystyle A}$, 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 ${\displaystyle r}$).

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

${\displaystyle \pi }$  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 ${\displaystyle r}$  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:

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

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

• Semicircle
• Sphere
• Squaring the circle
• Pi
• Tau

 

Wiki Commons has media related to Circle.

• Calculate the measures of a circle online