 What is a radius? The geometry of a circle

The radius of a circle is the distance from the center point to the edge of the circle. It’s the same distance anywhere on the circle, because the circle has radial symmetry.

So it doesn’t matter where you measure the radius on the circle, and if you know one radius measurement for a circle, then you know all of them. Try it for yourself and see! Each radius is a line segment reaching from the center of the circle to the perimeter.

## How do I get the radius from the circumference?

But suppose you *don’t* know the radius of this circle? Suppose all you know is that the circumference is 30 centimeters? Measuring the circumference of a tree.

One example would be if you were trying to measure a big tree. It would be pretty easy to take a rope and wrap it around the tree, and then measure the length of the rope to find out how big around the tree was (the circumference), but you’d have to cut the tree down to measure the radius.

## How do you find the radius of a circle?

We don’t want to cut this big tree down, so we’ll need to use math to figure out the radius instead. Luckily we know that the radius of any circle is always the same as half of the circumference divided by π (pi). So the radius of this tree is 30/π = 9.55 centimeters, divided by two is 4.77 centimeters.

## Prove the relationship between radius and circumference

How can we be sure that this is true for every circle? We can do a simple geometrical proof that will convince anybody that the circumference of a circle is always, always, always going to be half of the circumference divided by pi.

## When did people know how to calculate the radius from the circumference of a circle?

Mathematicians and engineers have known how to use approximate values of pi to calculate the radius of a circle certainly since about 2000 BC. In West Asia and in Egypt, mathematicians knew how to calculate the value of pi by around 1800 BC. They got it as accurate as 3.16. They used pi to figure out the area of circles and the volume of cylinders.

## What else is a radius good for?

Once you have the radius of a circle, you can use it to figure out the circumference of the circle – the distance around the outside – and the area of the circle. If you know the radius of a cylinder, you can use that and the height of the cylinder to calculate the volume of the cylinder. You can also calculate the volume of a cone from the radius. And if you know the radius of a sphere – a ball – you can use it to calculate the volume of the sphere and the surface area of the sphere.

So what is a radius? How do you find the radius of a circle? How do we know this is always true? Did you find out what you needed to know? Let us know in the comments!