Area of Circles & Composite Figures

Area measures the space inside a flat shape. For a rectangle you multiply length by width, and for a circle there is a formula that does the same job in one line: $A = \pi r^2$ . The $r$ is the radius, the distance from the center of the circle out to the edge.

Real figures are rarely a plain circle. A window is a rectangle with a half circle on top, and a track is two straight sides with curved ends. Those are composite figures. You handle them by cutting the shape into pieces you already know, finding each area, then putting the pieces back together.

The circle formula, and what gets squared

The formula is $A = \pi r^2$ . Square the radius first, then multiply by $\pi$ . That order matters: with a radius of $5$ , the area is $\pi \cdot 25$ , not $(\pi \cdot 5)^2$ and not $\pi \cdot 10$ . Squaring the radius is not the same as doubling it, and mixing those up is the single most common error in this topic.

Two answer styles show up. In terms of $\pi$ means you leave the symbol in place, so a radius of $3$ gives $9\pi$ , which is exact. If the problem says use $3.14$ for $\pi$ , multiply it out for a decimal. Area is always in square units, so a radius in centimeters gives an answer in square centimeters.

When the figure gives you the diameter

The diameter runs all the way across the circle through the center, so it is twice the radius. The area formula runs on the radius only. When a figure marks the diameter, halve it before anything else. A circle with a diameter of $12$ has a radius of $6$ , so its area is $\pi \cdot 6^2 = 36\pi$ .

Substituting the diameter straight into the formula is an expensive slip, because it gives four times the true area. Get in the habit of writing $r = \dfrac{d}{2}$ as your first line whenever a figure shows a line across the whole circle rather than one from the center to the edge.

Breaking a composite figure apart

Draw a line that splits the figure into familiar pieces, usually a rectangle and part of a circle. Find each area on its own, then add. A half circle is exactly half of a full circle, so its area is $\dfrac{1}{2}\pi r^2$ . A quarter circle is $\dfrac{1}{4}\pi r^2$ . The radius of that curved piece almost always comes from a side of the rectangle it sits against.

Some figures have a piece removed instead of attached, such as a circular hole punched out of a square plate. Same plan, different last step: find the whole shape, find the piece that was taken out, then subtract. Ask whether each piece is covered or empty before you decide to add or to subtract.

Worked examples

Example 1: area from the radius

A circle has a radius of $5$ units. Find its area. Use $3.14$ for $\pi$ .

Write the formula

A = \pi r^2

Substitute the radius

A = 3.14 \cdot 5^2

Square first, then multiply

A = 3.14 \cdot 25 = 78.5

Answer: $78.5$ square units.

Example 2: area from the diameter

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

Halve the diameter to get the radius

r = \dfrac{12}{2} = 6

Substitute into the formula

A = \pi \cdot 6^2

Square the radius and stop

A = 36\pi

Answer: $36\pi$ square inches.

Example 3: a rectangle with a half circle attached

A figure is a $10$ by $6$ rectangle with a half circle attached to one of the short sides. Find the total area. Use $3.14$ for $\pi$ .

Area of the rectangle

10 \cdot 6 = 60

The half circle spans the

6

side, so its radius is

3

r = \dfrac{6}{2} = 3

Area of the half circle

\dfrac{1}{2} \cdot 3.14 \cdot 3^2 = 14.13

Add the two pieces

60 + 14.13 = 74.13

Answer: $74.13$ square units.

Example 4: a piece removed

A square piece of metal measures $10$ centimeters on each side. A circle with a radius of $5$ centimeters is cut out of it. How much metal is left? Use $3.14$ for $\pi$ .

Area of the whole square

10 \cdot 10 = 100

Area of the circle removed

3.14 \cdot 5^2 = 78.5

Subtract the hole from the square

100 - 78.5 = 21.5

Answer: $21.5$ square centimeters of metal are left.

Try one yourself

Common questions

What is the difference between area and circumference?

Circumference measures the distance around a circle and uses $C = 2\pi r$ . Area measures the space inside and uses $A = \pi r^2$ . Circumference is a length, so its units are plain units; area is in square units. If the radius is squared in your work, you are finding area.

When should I leave

\pi

in my answer instead of using

3.14

Follow the instruction in the problem. In terms of $\pi$ means write $25\pi$ and leave it, which is the exact value. Use $3.14$ for $\pi$ means multiply it out to a decimal. Both are correct answers to different requests, so read that line before you compute.

How do I know the radius of the half circle in a composite figure?

The flat edge of the half circle is a side of the rectangle it is attached to, and that flat edge is the diameter. Halve it. If the half circle sits on a side of length $6$ , the radius is $3$ .

Want the video version?

Allday Everyday Math has video lessons, practice, and an AI tutor for every topic, Pre-Algebra through Algebra 2.

Try it for $1