# Area of Composite Shapes Involving Triangles

## Calculate areas of irregular shapes.

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Area of Composite Shapes Involving Triangles

Jerry was hired to paint the front of a neighbor’s shed. Using the illustration below for dimensions, how many square feet of paint does he need?

In this concept, you will learn to find the area of composite shapes.

### Finding the Area of Composite Shapes

The area of a rectangle is equal to its length times its width: \begin{align*}A=lw\end{align*}. The area of a triangle is equal to its base times its height divided by 2: \begin{align*}A=\frac{bh}{2}\end{align*}. In order to find the composite area of two or more shapes, simply find the area of each shape and add them together. The order in which you calculate the areas does not matter, and the commutative property states that it does not matter which order you add them in.

Let’s look at an example.

Find the area of the figure below.

First, if it’s not already done for you, divide the area into shapes you can work with.

Next, find the area of the rectangle.

\begin{align*}\begin{array}{rcl} A &=& lw \\ A &=& 9(5) \\ A &=& 45 \ in.^2 \end{array}\end{align*}

Then, find the area of one of the triangles.

In this figure, you will need to find the base first.

To find the base of one triangle, subtract 9 from 22, then divide by 2.

\begin{align*}\begin{array}{rcl} 22 - 9 &=& 13 \\ 13 \div 2 &=& 6.5 \end{array}\end{align*}

The base of each triangle is 6.5 inches.

Next, calculate the area of one triangle.

\begin{align*}\begin{array}{rcl} A &=& \frac{bh}{2} \\ A &=& \frac{(6.5)(5)}{2} \\ A &=& 16.25 \ in.^2 \end{array}\end{align*}

The last step is to add the three areas together. Remember, there are two triangles.

\begin{align*}\begin{array}{rcl} \text{Composite Area} & = & \text{Area of Triangle }1+ \text{Area of Rectangle} + \text{Area of Triangle }2 \\ A &=& A_{T1}+A_R+A_{T2} \\ A &=& 16.25+45 +16.25 \\ A &=& 77.5 \end{array}\end{align*}

The answer is 77.5 square inches.

### Examples

#### Example 1

Earlier, you given a problem about Jerry, who was hired to paint the front of a neighbor’s shed.

He needs to figure out how many square feet of paint he’ll use according to the following diagram.

First, calculate the area of the rectangle.

\begin{align*}\begin{array}{rcl} A_R &=& lw \\ A_R &=& 6 \times 7.5 \\ A_R &=& 45 \ sq \ ft \end{array}\end{align*}

The area of the rectangle is 45 square feet.

Next, calculate the area of the triangle.

\begin{align*}\begin{array}{rcl} A_T &=& \frac{bh}{2} \\ A_T &=& \frac{(6)(4)}{2} \\ A_T &=& 12 \ ft^2 \end{array}\end{align*}

\begin{align*}\begin{array}{rcl} A &=& A_R+A_T \\ A &=& 45 \ sq \ ft + 12 \ sq \ ft \\ A &=& 57 \ sq \ ft \end{array} \end{align*}

The answer is 57 square feet. Jerry needs a can of paint that will cover 57 sq. feet.

#### Example 2

A figure is made up of three triangles. Each triangle has a base of 6 inches and a height of 4 inches. What is the combined area of all three triangles?

First, use the formula to find the area of one triangle.

\begin{align*}\begin{array}{rcl} A &=& \frac{bh}{2} \\ A &=& \frac{(6)(4)}{2} \end{array} \end{align*}

Next, calculate.

\begin{align*}A=12 \ sq \ in\end{align*}

Then, remember that there are three triangles so multiply your answer times 3.

\begin{align*}\begin{array}{rcl} A_T &=& 12 \ sq \ in \times 3 \\ A_T &=& 77.5 \ sq \ in \end{array}\end{align*}

The answer is 77.5 square inches.

#### Example 3

A figure is made up of two triangles and one rectangle. Each triangle has a base of 5 inches and a height of 3 inches. The rectangle has a length of 4 inches and a width of 3 inches. What is the total area of the figure?

First, find the area of the rectangle.

\begin{align*}\begin{array}{rcl} A_R &=& lw \\ A_R &=& 4(3) \\ A_R &=& 12 \ in.^2 \end{array}\end{align*}

Next, find the area of one of the triangles.

\begin{align*}\begin{array}{rcl} A_T &=& \frac{bh}{2} \\ A_T &=& \frac{(5)(3)}{2} \\ A_T &=& 7.5 \ in.^2 \end{array}\end{align*}

Then, add the three areas together. Remember, there are two triangles.

\begin{align*}\begin{array}{rcl} A &=& A_{T1}+A_R+A_{T2} \\ A &=& 7.5+12+7.5 \\ A &=& 27 \end{array}\end{align*}

The answer is 27 square inches.

#### Example 4

A figure is made up of one triangle and one square. The square and the triangle have the same base length of 8 feet. The height of the triangle is 7 feet. What is the total area of the figure?

First, find the area of the square.

\begin{align*}\begin{array}{rcl} A_S &=& s^2 \\ A_S &=& 8^2 \\ A_S &=& 64 \ ft.^2 \end{array}\end{align*}

Next, find the area of the triangle.

\begin{align*}\begin{array}{rcl} A_T &=& \frac{bh}{2} \\ A_T &=& \frac{(8)(7)}{2} \\ A_T &=& 28 \ ft.^2 \end{array}\end{align*}

\begin{align*} \begin{array}{rcl} A &=& A_S+A_T \\ A &=& 64 \ sq \ ft+28 \ sq \ ft \\ A &=& 92 \ sq \ ft \end{array}\end{align*}

The answer is 92 square feet.

#### Example 5

Find the area of the composite figure below.

First, calculate the area of the rectangle.

\begin{align*}\begin{array}{rcl} A_R &=& lw \\ A_R &=& 17 \times 8.5 \\ A_R &=& 144.5 \ sq \ in \end{array}\end{align*}

The area of the rectangle is 144.5 square inches.

Next, calculate the area of the triangle. In order to do this, you first need to recognize that the height is equal to 17 inches minus 8.5 inches, or 8.5 inches.

Then, use the formula to find the area of the triangle.

\begin{align*}\begin{array}{rcl} A_T &=& \frac{bh}{2} \\ A_T &=& \frac{(17)(8.5)}{2} \\ A_T &=& 72.25 \ in^2 \end{array}\end{align*}

\begin{align*}\begin{array}{rcl} A &=& A_R+A_T \\ A &=& 144.5 \ sq \ in+72.25 \ sq \ in \\ A &=& 216.75 \ sq \ in \end{array} \end{align*}

The answer is 216.75 square inches.

### Review

Find the area of each combined figure.

1. A figure is made up of a triangle and a square. The square and the triangle have the same base of 7 inches. The triangle has a height of 5 inches, what is the total area of the figure?
2. A figure is made up of a triangle and a rectangle. The triangle has a height of 8 inches and a base of 9 inches. The rectangle has dimensions of \begin{align*}7 \ \text{inches} \times 9 \ \text{inches}\end{align*}. What is the area of the figure?
3. A figure is made up of four triangles. Each triangle has a base of 7 inches and a height of 9 inches. What is the total area of the figure?
4. A figure is made up of three triangles. Each triangle has a base of 5 feet and a height of 4.5 feet. What is the total area of the figure?
5. A figure is made up of two triangles and a square. The triangles and the square have the same base length of 5 feet. The triangles have a height of 4 feet. What is the total area of the figure?
6. A figure is made up of two triangles and a square. The triangles and the square have the same base length of 8 feet. The triangles have a height of 7 feet. What is the total area of the figure?
7. A figure is made up of one square and one triangle. The square has a side length of 9 feet. The triangle has a base of 7 feet and a height of 6 feet. What is the total area of the figure?
8. A figure is made up of two triangles. The triangles have the same base of 15 inches. One triangle has a height of 9 inches and one has a height of 11 inches. What is the total area of the figure?
9. A figure is made up of a triangle and a rectangle. The triangle has a height of 8.5 inches and a base of 10 inches. The rectangle has dimensions of \begin{align*}9 \ \text{inches} \times 10 \ \text{inches}\end{align*}. What is the area of the figure?
10. A figure is made up of a triangle and a rectangle. The triangle has a height of 11 inches and a base of 9.5 inches. The rectangle has dimensions of \begin{align*}12 \ \text{inches} \times 14 \ \text{inches}\end{align*}. What is the area of the figure?

Solve each problem.

1. Julius drew a triangle that had a base of 15 inches and a height of 11 inches. What is the area of the triangle Julius drew?
2. A triangle has an area of 108 square centimeters. If its height is 9 cm, what is its base?
3. What is the height of a triangle whose base is 36 inches and area is 234 square inches?
4. Tina is painting a triangular sign. The height of the sign is 32 feet. The base is 27 feet. How many square feet will Tracy paint?
5. Tina drew a picture of a triangle with a base of 6 inches and a height of 5.5 inches. What is the area of the triangle?

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### Vocabulary Language: English

TermDefinition
Area Area is the space within the perimeter of a two-dimensional figure.
Base The side of a triangle parallel with the bottom edge of the paper or screen is commonly called the base. The base of an isosceles triangle is the non-congruent side in the triangle.
Height The height of a triangle is the perpendicular distance from the base of the triangle to the opposite vertex of the triangle.
Triangle A triangle is a polygon with three sides and three angles.