# 9.19: Areas of Combined Figures Involving Circles

**At Grade**Created by: CK-12

Have you ever tried to figure out the area of a figure that was made up of two figures and not just one?

Marcy has a picture of a figure that is made up of a square and half of a circle or a semi - circle. The square has a side length of 8 inches. The circle's diameter is the same as the side length of the square.

What is the area of the figure?

**This Concept will show you how to figure out the area of combined figures involving circles. Pay attention and we will revisit this dilemma at the end of the Concept.**

### Guidance

Sometimes we may be asked to find the area of a combined figure. Combined figures often include portions of circles, such as a quarter or semicircle (which is a half circle). We can find the area of combined figures by breaking them down into smaller shapes and finding the area of each piece.

We can calculate the area of a portion of a circle. As long as we know the radius of the circle, we can find its area. Then we can divide that area into smaller pieces or subtract a portion to find the area of part of the circle. Let’s try this out.

What is the area of the figure below?

This figure is a semicircle, or half of a circle. Remember that a diameter always divides a circle in half. Therefore the edge measuring 17 inches is the circle’s diameter. Can we use it to find the area of the whole circle?

We sure can! The radius of the circle must be \begin{align*}17 \div 2 = 8.5\end{align*} inches. Now let’s use the formula to solve for area.

\begin{align*}A & = \pi r^2\\ A & = \pi (8.5)^2\\ A & = 72.25 \ \pi\\ A & = 226.87 \ in.^2\end{align*}

**We know that a whole circle with a radius of 8.5 inches (and a diameter of 17 inches) is 226.87 square inches. Therefore the semicircle figure has an area of \begin{align*}226.87 \div 2 = 113.44\end{align*} square inches.** As long as we can find the area of a whole circle, we can divide or subtract to find the area of a portion of a circle.

Now let’s look at a combined figure.

Find the area of the figure below.

First, we have to find the area of the rectangle. We can do this by multiplying the length times the width. Then we can find the area of the circle. If you notice, the width of the rectangle is also the diameter of the circle. This will help us when we want to find the area of the circle.

Let’s start with the rectangle.

\begin{align*}A&=lw\\ A&=6(8)\\ A&=48\end{align*}

**The area is 48 square inches for the rectangle.**

Now let’s look at the semi-circle. If the diameter is the width which is 6 inches, then the radius is 3 inches. We can find the area of a circle now.

\begin{align*}A&= \pi r^2 \\ A& =3.14(3^2) \\ A& =3.14(9) \\ A& =28.26 \ sq.inches \end{align*}

Now this is the area of a whole circle. We only need the area of a semi-circle. Let’s divide this value in half.

The area of the semi-circle is 14.13 square inches.

**Now we add the two areas together.**

**48 +14.13 = 62.13 square inches**

**The area of the entire figure is 62.13 square inches.**

Now it's your turn to try a few. Figure out each area.

#### Example A

Two semi - circles with a square in the middle. The square has a side length of 6 inches. The semi - circles have the same diameter as the side length.

**Solution: \begin{align*}149.04 \ in^2\end{align*}**

#### Example B

A rectangle with a length of 5 feet and a width of 3 feet connected to a semi - circle with the same diameter as the width.

**Solution: \begin{align*}29.13 \ ft^2\end{align*}**

#### Example C

A square with a side length of 4 mm and a semi - circle with a diameter of the same side length as the square.

**Solution: \begin{align*}41.12 \ mm^2\end{align*}**

Here is the original problem once again.

Marcy has a picture of a figure that is made up of a square and half of a circle or a semi - circle. The square has a side length of 8 inches. The circle's diameter is the same as the side length of the square.

What is the area of the figure?

To figure this out, we can first figure out the area of the square.

\begin{align*}A = s^2\end{align*}

\begin{align*}A = 8^2\end{align*}

\begin{align*}A = 64 \ in^2\end{align*}

**This is the area of the square.**

Next, we have to figure out the area of half of a circle. Let's figure out the area of the whole circle first.

\begin{align*}A = \pi r^2\end{align*}

\begin{align*}A = \pi 8^2\end{align*}

\begin{align*}A = 200.96 \ in^2\end{align*}

This is the area of the whole circle. We only need half of this area, so we divide this number by 2.

\begin{align*}200.96 \div 2 = 100.48\end{align*}

Now, we add the two areas together.

\begin{align*}100.48 + 64 = 164.48 \ in^2\end{align*}

**This is our answer.**

### Vocabulary

Here are the vocabulary words in this Concept.

- Circle
- a set of connected points that are equidistant from a center point.

- Diameter
- the distance across the center of a circle.

- Radius
- the distance from the center of the circle to the outer edge.

- Area
- the space inside a two-dimensional figure

### Guided Practice

Here is one for you to try on your own.

Johann drew a design for a new sign for the cafe at school. The sign is made up of a square and a semi - circle. The square has a side length of 9 inches. The semi - circle has the same diameter as the square.

What is the area of the sign?

**Answer**

First, we have to figure out two different measurements for area and add them together. Let's start with the square.

\begin{align*}A = s^2\end{align*}

\begin{align*}A = 9^2\end{align*}

\begin{align*}A = 81 \ in^2\end{align*}

Next, we figure out the area of the half circle. Look at how the formula for the area of a circle can be adjusted.

\begin{align*}A = \frac{1}{2}[\pi r^2]\end{align*}

\begin{align*}A = \frac{1}{2}[3.14(9^2)]\end{align*}

\begin{align*}A = \frac{1}{2}(254.34)\end{align*}

\begin{align*}A = 127.17 \ in^2\end{align*}

This is the area of half of the circle.

Now we add these areas together.

\begin{align*}127.17 + 81 = 208.17 \ in^2\end{align*}

**This is our answer.**

### Video Review

Here is a video for review.

- This is a James Sousa video about finding the area of a combined figure.

### Practice

Directions: Find the area of each combined figure.

1. A square and a semi - circle. The square has a side length of 11 mm. The diameter of the circle matches the square's side.

2. A square and a semi - circle. The square has a side length of 8.5 inches. The diameter of the circle matches the square's side.

3. A square and a semi - circle. The square has a side length of 7.25 inches. The diameter of the circle matches the square's side.

4. A square and a semi - circle. The square has a side length of 13 feet. The diameter of the circle matches the square's side.

5. A square and a semi - circle. The square has a side length of 15.5 feet. The diameter of the circle matches the square's side.

6. A rectangle and a semi - circle. The rectangle has a length of 8 feet and a width of 5 feet. The diameter of the circle matches the width.

7. A rectangle and a semi - circle. The rectangle has a length of 8.5 feet and a width of 6 feet. The diameter of the circle matches the width.

8. A rectangle and a semi - circle. The rectangle has a length of 9 inches and a width of 4.5 inches. The diameter of the circle matches the length.

9. A rectangle and a semi - circle. The rectangle has a length of 7 feet and a width of 4 feet. The diameter of the circle matches the length.

10. A rectangle and a semi - circle. The rectangle has a length of 5.5 feet and a width of 3.5 feet. The diameter of the circle matches the width.

11. A triangle and a semi - circle. The triangle has a base of 5 inches and a height of 4 inches. The diameter of the circle matches the base of the triangle.

12. A triangle and a semi - circle. The triangle has a base of 7 inches and a height of 6 inches. The diameter of the circle matches the base of the triangle.

13. A triangle and a semi - circle. The triangle has a base of 5.5 inches and a height of 4 inches. The diameter of the circle matches the base of the triangle.

Directions: Solve each problem.

14. Rob is painting large polka dots on a sheet for the backdrop of the school musical. He painted 16 polka dots, each with a radius of 3 feet. What is the total area that the polka dots cover?

15. The librarian is having the library at her school carpeted. The library is a circular room with a diameter of 420 feet. How many square feet of carpet will she need to order?

### Image Attributions

Here you'll learn to find areas of combined figures involving circles.