Math Core

How much pizza is in a pizza? How much grass does a round sprinkler water? Those questions ask for the area of a circle: the amount of flat space inside it. Like circumference, the area formula uses $\pi$ .

Where the formula comes from

Cut a circle into many thin wedges, like pizza slices. Line them up, pointing up and down in turn, so they fit together.

Half the wedges are on top and half are on the bottom, so the bumpy top and bottom each have length equal to half the circumference: $\dfrac{1}{2} \times 2\pi r = \pi r$ .
The height of the shape is the length of a wedge, which is the radius $r$ .

The more wedges you use, the more the shape looks like a rectangle with base $\pi r$ and height $r$ . Its area is $\pi r \times r = \pi r^2$ . Cutting and moving pieces doesn't change the area, so that's the circle's area too.

Area of a circle

A = \pi r^2

Square the radius first, then multiply by $\pi$ . Area is measured in square units.

A circle with radius 4 cm.

Worked example: Area from the radius

Find the area of the circle above in terms of $\pi$ , then approximate it using $3.14$ .

A = \pi r^2 = \pi \times 4^2 = 16\pi \approx 16 \times 3.14 = 50.24 \text{ cm}^2

Worked example: Area from the diameter

A round table has diameter $10$ feet. Find its area using $3.14$ for $\pi$ .

The radius is $10 \div 2 = 5$ feet, so

A = \pi \times 5^2 = 25\pi \approx 25 \times 3.14 = 78.5 \text{ ft}^2.

Common mistake

Two mistakes show up all the time:

Using the diameter in $\pi r^2$ . Always cut the diameter in half first.
Doubling instead of squaring. For $r = 4$ , $r^2 = 4 \times 4 = 16$ , not $4 \times 2 = 8$ .

Area from circumference

Circumference and area both depend on the radius, so the radius connects them. If you know one, find $r$ , then use it to find the other.

Worked example: From circumference to area

A circle has circumference $12\pi$ inches. Find its area in terms of $\pi$ .

$2\pi r = 12\pi$ , so $r = 6$ inches. Then $A = \pi \times 6^2 = 36\pi$ square inches.

Shapes made from circles

Many real shapes combine circles with rectangles or other figures. Break the shape into parts you know, find each area, and add (or subtract).

A window: a rectangle with a semicircle on top.

Worked example: A window

The window above is a rectangle $8$ ft wide and $4$ ft tall, topped by a semicircle. Find its area using $3.14$ for $\pi$ .

Rectangle: $8 \times 4 = 32 \text{ ft}^2$ .
Semicircle: its diameter is the top of the rectangle, $8$ ft, so its radius is $4$ ft. The area is half a circle: $\dfrac{1}{2} \times \pi \times 4^2 = 8\pi \approx 25.12 \text{ ft}^2$ .

Total: $32 + 25.12 = 57.12 \text{ ft}^2$ .

Tip

Keep answers in terms of $\pi$ until the very end. It's exact, and it makes the arithmetic easier: $32 + 8\pi$ is simpler to work with than a long decimal.

Practice

Practice 1

A circle has radius $3$ cm. What is its area in square centimeters? Give your answer in terms of $\pi$ .