What if you drew a basic house with a triangle on top of a square? How could you find the area of this composite shape? After completing this Concept, you'll be able to calculate the area of irregular shapes that are made up of two or more shapes you already know.

### Watch This

Area of Composite Shapes CK-12

### Guidance

**Perimeter** is the distance around a shape. The perimeter of any figure must have a unit of measurement attached to it. If no specific units are given (feet, inches, centimeters, etc), write “units.”

**Area** is the amount of space inside a figure. If two figures are congruent, they have the same area (**Congruent Areas Postulate**).

A **composite shape** is a shape made up of other shapes. To find the area of such a shape, simply find the area of each part and add them up.

**Area Addition Postulate:** If a figure is composed of two or more parts that do not overlap each other, then the area of the figure is the sum of the areas of the parts.

#### Example A

Find the area of the figure below.

Divide the figure into a triangle and a rectangle with a small rectangle cut out of the lower right-hand corner.

\begin{align*}A &= A_{top \ triangle}+A_{rectangle}-A_{small \ triangle}\\ A &= \left(\frac{1}{2} \cdot 6 \cdot 9\right)+(9 \cdot 15)\left) - (\frac{1}{2} \cdot 3 \cdot 6\right)\\ A &= 27+135-9\\ A &= 153 \ units^2\end{align*}

#### Example B

Divide the shape into two rectangles and one triangle. Find the area of the two rectangles and triangle:

Rectangle #1: \begin{align*}\text{Area }= 24(9+12)=504 \ units^2\end{align*}

Rectangle #2: \begin{align*}\text{Area }=15(9+12)=315 \ units^2\end{align*}

Triangle: \begin{align*}\text{Area }=\frac{15(9)}{2}=67.5 \ units^2\end{align*}

#### Example C

Find the area of the entire shape from Example B (you will need to subtract the area of the small triangle in the lower right-hand corner).

\begin{align*}\text{Total Area }=504+315+67.5-\frac{15(12)}{2}=796.5 \ units^2\end{align*}

Area of Composite Shapes CK-12

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### Guided Practice

- Divide the shape into two triangles and one rectangle.
- Find the area of the two triangles and rectangle.
- Find the area of the entire shape.

**Answers**

1. One triangle on the top and one on the right. Rectangle is the rest.

2. Area of triangle on top is \begin{align*}\frac{8(5)}{2}=20 \ units^2\end{align*}. Area of triangle on right is \begin{align*}\frac{5(5)}{2}=12.5 \ units^2\end{align*}. Area of rectangle is \begin{align*}375 \ units^2\end{align*}.

3. Total area is \begin{align*}407.5 \ units^2\end{align*}.

### Explore More

Use the picture below for questions 1-4. Both figures are squares.

- Find the area of the outer square.
- Find the area of one grey triangle.
- Find the area of all four grey triangles.
- Find the area of the inner square.

Find the areas of the figures below. You may assume all sides are perpendicular.

Find the areas of the composite figures.

Use the figure to answer the questions.

- What is the area of the square?
- What is the area of the triangle on the left?
- What is the area of the composite figure?