# Area of a Circle

## 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,

*, times the square of the circle's radius gives you the area of the inside of the circle, in square units.*

**π**### 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*,

*, and so on.*

**square feet**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,

*, in the same units.*

**r**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

A

**406-mm**bicycle wheelThe London Eye Ferris wheel with a radius of

**60 meters**A

**26-inch**bicycle wheelThe world's largest pizza had a radius of

**61 feet**,**4 inches**(**736 inches**)

### Answers

A

**406-mm**bicycle wheel has a**radius**of**203 mm**:$A=\pi \times 203m{m}^{2}$

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

The London Eye Ferris wheel's

**60 meter radius:**$A=\pi \times 60{m}^{2}$

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

A

**26-inch**bicycle wheel has a radius,, of**r****13 inches**:$A=\pi \times 13i{n}^{2}$

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

The world's largest pizza with its

**736 inch radius**:$A=\pi \times 736i{n}^{2}$

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

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!