# Radius, diameter, & circumference

CCSS Math: 7.G.B.4

Learn the relationship between the radius, diameter, and circumference of a circle.

## What is a circle?

We've all seen circles before. They have this perfectly round shape, which makes them perfect for hoola-hooping!

Every circle has a center, which is a point that lies exactly at the... well... center of the circle.
A circle is a shape where distance from the center to the edge of the circle is always the same:

You might have suspected this before, but in fact, the distance from the center of a circle to any point on the circle itself is exactly the same.

## Radius of a circle

This distance is called the radius of the circle.

## Diameter of a circle

The diameter is the length of the line through the center that touches two points on the edge of the circle.

Notice that a diameter is really just made up of two radii (by the way, "radii" is just the plural form of radius):

So, the diameter $d$ of a circle is twice the radius $r$:

## Circumference of a circle

The circumference is the distance around a circle (its perimeter!):

Here are two circles with their circumference and diameter labeled:

Let's look at the ratio of the circumference to diameter of each circle:

Circle 1 | Circle 2 | |
---|---|---|

$\dfrac{\text{Circumference}}{\text{Diameter}}$: | $\dfrac{3.14159...}{1} = \redD{3.14159...}$ | $\dfrac{6.28318...}{2} = \redD{3.14159...}$ |

Fascinating! The ratio of the circumference $C$ to diameter $d$ of both circles is $\redD{3.14159...}$

This turns out to be true for all circles, which makes the number $\redD{3.14159...}$ one of the most important numbers in all of math! We call the number pi (pronounced like the dessert!) and give it its own symbol $\redD\pi$.

Multiplying both sides of the formula by $d$ gives us

which lets us find the circumference $C$ of any circle as long as we know the diameter $d$.

## Using the formula $C = \pi d$

Let's find the circumference of the following circle:

The diameter is $10$, so we can plug $d = 10$ into the formula $C = \pi d$:

That's it! We can just leave our answer like that in terms of $\pi$. So, the circumference of the circle is $10 \pi$ units.

Your turn to give it a try!