Radius of a circle

Written by

Malcolm McKinsey

January 16, 2023

Fact-checked by

Paul Mazzola

Radius of a circle

The radius of a circle is a line segment with one end at the center of the circle, and the other end on the circle itself. The radius is identified with a lower-case, italic r.

Radius of a circle formula

The formula you use depends on what is known about the circle. Below are three different formulas you can use to find the radius, r, of a circle.

• If given circumference (C), then the formula is: $r=\frac{C}{2\pi }$

• If given area (A), then the formula is: $r=\sqrt{\frac{A}{\pi }}$

• If given diameter (d), then the formula is: $r=\frac{d}{2}$

The relationship between radius and diameter is an important one to know when learning to how to calculate the radius.

Since the radius is a line segment from the center to the circle, and the diameter, d, is a line segment from on side of a circle through the center of a circle and out to the other side of the circle, it follows that a radius is $\frac{1}{2}$ a diameter.

How to find the radius of a circle

To find the radius of a circle, plug the given information into the correct formula. You can find the radius of any circle if you are given the circumference, area, or diameter.

How to find radius from circumference

No matter the size of the circle, the relationship of the radius to the circumference, C, is a constant, giving us the formula to find radius from circumference:

You can also calculate the circumference of a circle with a given radius by using algebra to isolate the C in our formula.

The radius to circumference formula is: $C=2\pi r$

Radius of circle from area

You can use the area to find the radius and the radius to find the area of a circle. The area of a circle is the space it occupies, measured in square units.

Given the area, A, of a circle, its radius is the square root of the area divided by pi:

The formula for radius to area is: $A=\pi {r}^{2}$

How to find radius from diameter

If you know the diameter of a circle, you can find its radius by dividing the diameter by 2:

This is the simplest formula you can use to get the radius.

If you need to go from radius to diameter, multiply radius times 2: d = 2r

Radius of a circle examples

1. A circle has a diameter of 6 inches. What is its radius?

2. A circle has a radius of 98 feet. What is its diameter?

3. A circle with a circumference of 365 meters will have a radius of how many meters?

4. A circle with a radius of 11 inches will have a circumference of how many inches?

5. With a radius of 41 mm, a circle will have an area equal to, what?

We don't like to talk in circles, but make sure you work these out before checking the answers below!

1. The radius of a 6 inch circle is 3 inches.

2. A circle has a radius of 98 feet, so its diameter must be 196 feet.

3. A circle with a circumference of 365 meters will have a radius of 58.4 meters.

4. A circle with a radius of 11 inches will have a circumference of 69.12 inches.

5. With a radius of 41 mm, a circle will have an area of roughly 5,281 square millimeters.