# 6.8: Area of Shaded and Composite Figures

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

## Learning Objectives

- Use formulas to find the area of specific types of two-dimensional shapes by analyzing them as the sum or difference of smaller polygons.

## Congruent Areas

Before we find the area of more complicated figures, we must know that:

*If two figures are congruent, they have the same area.*

This means that:

If two shapes are the same, the area inside them is also _____________________.

This is obvious because congruent figures have the same amount of space inside them. However, two figures with the same area are not necessarily congruent.

## Area of a Whole is the Sum of Parts

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.*

This means that:

You can break down a figure into parts and ___________ the areas of those parts together to get the area of the whole figure.

This is the familiar idea that a whole is the sum of its parts. In practical problems you may find it helpful to break a figure down into parts.

**Reading Check**

1. *True or false: If two shapes are congruent, then they have the same area*.

2. *True or false: If two shapes have the same area, then they are congruent*.

3. *In your own words, describe what the phrase means:*

“A whole is the sum of its parts.”

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**Example 1**

*Find the area of the figure below.*

Luckily, you do not have to learn a special formula for an irregular pentagon (which this figure is because its sides are not congruent.)

Instead, you can break the figure down into a trapezoid and a triangle like the dotted line does below. Use the area formulas for these parts to find the area of the whole figure.

The shape created on the *left* of the dotted line is a _______________________.

The shape created on the *right* of the dotted line is a _______________________.

Without numbers, we can review our formulas in describing the steps:

To find the area of the trapezoid, use the formula: \begin{align*}A = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}

To find the area of the triangle, use the formula: \begin{align*}A = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}.

Because the area of the whole is the sum of its parts, we simply add together the areas of the trapezoid and the triangle to find the area of the entire figure!

**Example 2**

*What is the area of the figure shown below?*

Break the figure down into two rectangles:

First, look at the *top rectangle*:

The larger rectangle has a base of 45 cm and a height of 22 cm.

Since the area of a rectangle is \begin{align*}\text{base} \cdot \text{height}\end{align*},

\begin{align*}A = \underline{\;\;\;\;\;\;\;\;\;\;\;\;} \ cm \cdot \underline{\;\;\;\;\;\;\;\;\;\;\;\;} \ cm = 990 \ cm^2\end{align*}

Second, look at the *bottom rectangle:*

The smaller rectangle has a base of 20 cm and a height of 8 cm.

\begin{align*}A = \underline{\;\;\;\;\;\;\;\;\;\;\;\;} \ cm \cdot \underline{\;\;\;\;\;\;\;\;\;\;\;\;} \ cm = 160 \ cm^2\end{align*}

Area of the whole figure is the sum of its parts, so:

\begin{align*}\text{Area} & = \text{area of top rectangle} + \text{area of bottom rectangle}\\ A & = \underline{\;\;\;\;\;\;\;\;\;\;\;\;} \ cm^2 + \underline{\;\;\;\;\;\;\;\;\;\;\;\;} \ cm^2 = 1150 \ cm^2\end{align*}

The area of the entire figure is \begin{align*}1150 \ cm^2\end{align*}.

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