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Areas of Combined Figures Involving Circles

Calculate areas of irregular shapes.

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Areas of Combined Figures Involving Circles
License: CC BY-NC 3.0
 

Fred’s hobby is wood working, and today he is building a dollhouse for his daughter. The house will include two front bay windows with the dimensions below. Fred needs to know the area of the window, so he can have the plastic windows cut at the crafts store.

License: CC BY-NC 3.0

How can Fred determine the area of the window?

In this concept, you will learn to find areas of combined figures involving parts of circles.

Finding Areas of Combined Figures

Sometimes, there will be figures that are not quadrilaterals or circles, but are combined figures. A combined figure is a figure that is made up of more than one type of polygon. You can still figure out the area of combined figures, by finding the area of each of the figures and then combing them to get the total area.

Let’s look at an example.

Take a look at the figure below. It is a half circle and a rectangle combined into one figure.

What is the area of the figure?

License: CC BY-NC 3.0

To solve this problem, you first have to look at which figures have been combined. Here you have one-half of a circle and a rectangle.

You will need to figure out the area of the rectangle, the area of half of the circle, and then add the two areas. This will give you the area of the combined figure.

Let’s start with the rectangle.

\begin{align*}A = lw\end{align*}

The length of the rectangle is 6 inches. The width of the rectangle is 3 inches.

First, substitute the given information into the equation to find the area of the rectangle.

\begin{align*}\begin{array}{rcl} A&=& (6)(3) \\ A&=& 18 \ in^2 \\ \end{array}\end{align*}

The rectangle’s area is \begin{align*}18 \ in^2\end{align*}.

Next, find the area of the circle.

Notice that the length of the rectangle is also the diameter of the circle, which is 6 inches. If the diameter of the circle is 6 inches, then the radius is 3 inches. Remember that the radius is one-half of the diameter.

The goal is to figure out the area of one whole circle and then divide that in half for the area of half of the circle.

First, substitute the given information into the equation for the area of a circle.

\begin{align*}\begin{array}{rcl} A&=& \pi r^2 \\ A&=& (3.14)(3 ^2) \\ A&=& 28.26 \ in^2 \end{array}\end{align*}

This is the area of the whole circle.

Next, because the figure is only half of a circle, divide this in two.

\begin{align*}28.26 \div 2 = 14.13 \ in\end{align*}

This is the area of the half circle.

Then, combine the area of the rectangle with the area of the half circle. This will equal the area of the entire figure.

\begin{align*}18 + 14.13= 32.13\end{align*}

The area of the combined figure is \begin{align*}32.13 \ in^2\end{align*}.

Examples

Example 1

Earlier, you were given a problem about Fred and the dollhouse he’s building for his daughter.

Fred designed two bay windows that were a combination of a rectangle and a half circle. The windows have the following dimensions.

License: CC BY-NC 3.0

Fred knows that the total area of the window will be the combined area of the rectangle and the half circle, so he decides to find the area of each and then combine them.

First, Fred finds the area of the rectangle by substituting the given information into the equation.

\begin{align*}A= lw\end{align*}The length is 6 inches and the width is 8 inches.

\begin{align*}A = (6)(8)= 48 \ \text{sq}.\text{inches}\end{align*}

The area of the rectangle is 48 square inches.

Next, this is a whole circle divided in half, so Fred finds the area of the whole circle by substituting the given information into the equation. Note that the radius is 4, half of the 8-inch diameter of the circle.

\begin{align*}\begin{array}{rcl} A&=&(3.14)r^2 \\ A&=&(3.14)4^2 \\ A&=&(3.14)(16) \\ A&=& 50.24 \ \text{sq}.\text{inches} \end{array}\end{align*}

Then, because this is the area of the whole circle, divide it by 2.

\begin{align*}50.24 \div 2 = 25.12 \ \text{sq}.\text{inches}\end{align*} 

The area of the half circle is 25.12 square inches.

Finally, add the area of the rectangle with the area of the half circle.

\begin{align*}48+ 25.12 = 73.12 \ \text{sq}.\text{inches}\end{align*}

The area of the combined figure is 73.12 square inches. So, each bay window has an area of 73.12 square inches.

Example 2

Find the area of the combined figure below.

License: CC BY-NC 3.0

First, find the area of the rectangle by substituting the given information into the equation.

\begin{align*}A=lw\end{align*}

The length is 24 ft. and the width is 8 feet.

\begin{align*}A = (8)(24) = 192 \ \text{sq}. \text{feet}\end{align*}

The area of the rectangle is \begin{align*}192\ \text{sq}. \text{feet}\end{align*}.

Next, find the area of the circle as if it were a whole circle by substituting the given information into the equation.

\begin{align*}\begin{array}{rcl} A&=&(3.14)r^2 \\ A&=&(3.14)12^2 \\ A&=&(3.14)(144) \\ A&=& 452.16 \ \text{sq}.\text{feet} \end{array}\end{align*}

Then, divide the area of the whole circle in half.

\begin{align*}A= 226.08 \ \text{sq}.\text{feet}\end{align*}

Finally, add the area of the rectangle with the area of half of the circle.

\begin{align*}192+226.08 = 418.08 \ \text{sq}.\text{feet}\end{align*}

The area of the combined figure is 418.08 sq. feet.

Example 3

Find the area of the combined figure. Remember, separate the figure and find the area of the parts, then combine the areas.

License: CC BY-NC 3.0

First, find the area of the square in the middle by substituting the given information into the equation.

\begin{align*}A=lw\end{align*}

The length is 6 inches and the width is 6 inches.

\begin{align*}A= (6)(6) = 36 \ \text{sq}.\text{inches}\end{align*}

The area of the square is 36 square inches.

Next, this is a whole circle divided into two by the square, so find the area of the whole circle by substituting the given information into the equation. Note that the radius is 3 in.

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

This is the combined area of both halves of the circle.

Finally, add the area of the rectangle with the area of the circle.

\begin{align*}36 + 28.26 = 64.26 \ \text{sq}.\text{inches}\end{align*}

The area of the combined figure is 64.26 square inches.

Example 4

Answer the following question.

Can you figure out the area of a figure made up of two congruent circles? How?

First, you could figure out the area of both circles using the given information.

Next, you would add the areas of both circles to get the area of the combined figure.

Example 5

Find the area of the figure.

License: CC BY-NC 3.0

This is a half circle.

First, note that the radius is half of the 17-inch diameter. So the radius is 8.5 inches.

Next, substitute this information into the area of a circle equation.

\begin{align*}\begin{array}{rcl} A&=&(3.14)r^2 \\ A&=&(3.14)8.5^2 \\ A&=&(3.14)(72.25) \\ A&=& 226.87 \ \text{sq}.\text{inches} \end{array}\end{align*}

This is the area of the whole circle.

Then, divide the area of the whole circle by 2 to get the area of the figure.

\begin{align*}226.87 \div 2 = 113.43 \ \text{sq}.\text{inches}\end{align*} 

The area of the figure is 113.43 square inches.

Review

Use each image to answer the following questions.

License: CC BY-NC 3.0

  1. Which two figures are pictured here?
  2. What is the area of the rectangle?
  3. What is the area of the circle if it were a whole circle?
  4. What is the area of half of the circle?
  5. What is the area of the whole figure?

License: CC BY-NC 3.0

  1. What is the radius of this circle?
  2. What is the diameter of this circle?
  3. What is the circumference of this circle?
  4. What is the area of the circle?
  5. If this circle were half of a circle, what would the new area be?

License: CC BY-NC 3.0

  1. What is the radius of this circle?
  2. What is the diameter of this circle?
  3. What is the circumference of this circle?
  4. What is the area of the circle?
  5. If this circle were half of a circle, what would the new area be?

Review (Answers)

To see the Review answers, open this PDF file and look for section 10.10. 

Resources

 

 

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Vocabulary

\pi

\pi (Pi) is the ratio of the circumference of a circle to its diameter. It is an irrational number that is approximately equal to 3.14.

Area

Area is the space within the perimeter of a two-dimensional figure.

Diameter

Diameter is the measure of the distance across the center of a circle. The diameter is equal to twice the measure of the radius.

Pi

\pi (Pi) is the ratio of the circumference of a circle to its diameter. It is an irrational number that is approximately equal to 3.14.

Radius

The radius of a circle is the distance from the center of the circle to the edge of the circle.

Squaring

Squaring a number is multiplying the number by itself. The exponent 2 is used to show squaring.

Image Attributions

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