Area of a Circle

Written by

Malcolm McKinsey

January 16, 2023

Fact-checked by

Paul Mazzola

What is the area of a circle?

A circle is not a square, but a circle's area (the amount of interior space enclosed by the circle) is measured in square units. Finding the area of a square is easy: length times width.

A circle, though, has only a diameter, or distance across. It has no clearly visible length and width, since a circle (by definition) is the set of all points equidistant from a given point at the center.

Yet, with just the diameter, or half the diameter (the radius), or even only the circumference (the distance around), you can calculate the area of any circle.

How to find the area of a circle

Recall that the relationship between the circumference of a circle and its diameter is always the same ratio, 3.14159265, pi, or π. That number, π, times the square of the circle's radius gives you the area of the inside of the circle, in square units.

Area of a circle formula

If you know the radius, r, in whatever measurement units (mm, cm, m, inches, feet, and so on), use the formula $\pi {r}^{2}$ to find area, A:

The answer will be square units of the linear units, such as $m{m}^{2}$, $c{m}^{2}$, ${m}^{2}$, square inches, square feet, and so on.

Say we have a circle with a radius of 7 meters. What is its area?

Area of a circle using diameter

If you know the diameter, d, in whatever measurement units, take half the diameter to get the radius, r, in the same units.

Here is the real estate development of Sun City, Arizona, a circular town with a diameter of 1.07 kilometers. What is the area of Sun City?

First, find half the diameter, given, to get the radius:

Plug in the radius into our formula:

To convert square meters, ${m}^{2}$, to square kilometers, $k{m}^{2}$, divide by 1,000,000:

Sun City's westernmost circular housing development has an area of nearly 1 square kilometer!

How to calculate the area of a circle

Try these area calculations for four different circles. Be careful; some give the radius, r, and some give the diameter, d.

Remember to take half the diameter to find the radius before squaring the radius and multiplying by π.

Problems

1. A 406-mm bicycle wheel

2. The London Eye Ferris wheel with a radius of 60 meters

3. A 26-inch bicycle wheel

4. The world's largest pizza had a radius of 61 feet, 4 inches (736 inches)

Answers

1. A 406-mm bicycle wheel has a radius of 203 mm:

   1. $A=\pi \times 203m{m}^{2}$

   2. $A=637.7433m{m}^{2}$

2. The London Eye Ferris wheel's 60 meter radius:

   1. $A=\pi \times 60{m}^{2}$

   2. $A=188.4955{m}^{2}$

3. A 26-inch bicycle wheel has a radius, r, of 13 inches:

   1. $A=\pi \times 13i{n}^{2}$

   2. $A=530.9291i{n}^{2}$

4. The world's largest pizza with its 736 inch radius:

   1. $A=\pi \times 736i{n}^{2}$

   2. $A=\mathrm{1,701,788.17}i{n}^{2}$

That is $\mathrm{11,817.97}f{t}^{2}$ of pizza! Yum! Anyway, how did you do on the four problems?

Area of a circle using circumference

If you have no idea what the radius or diameter is, but you know the circumference of the circle, C, you can still find the area.

Area and circumference formula

Circumference (the distance around the circle) is found with this formula:

That means we can take the circumference formula and "solve for r," which gives us:

We can replace r in our original formula with that new expression:

That expression simplifies to this:

That formula works every time!

How to find the area with circumference

Think of a beautiful, reasonable-sized pizza you and three friends can share. You happen to know the circumference of your pizza is 50.2655 inches, but you do not know its total area. You want to know how many square inches of pizza you will each enjoy. Substitute 50.2655 inches for C in the formula:

Equally divide that total area for a full-sized pizza among four friends, and you each get $50.2655i{n}^{2}$ of pizza! That's about a third of a square foot for each of you! Yum, yum!