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 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 $\pi$ (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 ($\pi$ 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 $3.1415926536$.

	$\pi =\frac{c}{d}$
$\therefore \text{(therefore)}$	$c=2\pi r$
	$c=\pi d$

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

Radius

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

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

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

Related pages

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

Other websites

• Calculate the measures of a circle online