## Areas of Complex Shapes

## Introduction

Many polygons have areas that can be found using formulas. When a figure is comprised of two or more individual polygons, the area can be found by adding up each of the individual areas.

The polygon above is comprised of two rectangles and its area is simply the sum of the areas of the individual rectangles. As long as you know how to find the area of each individual shape that comprises the complex figure, finding the total area is simply a matter of finding each individual area and adding the areas up.

## Lesson

It makes sense to get your feet wet by starting with a simple problem, then gradually learning more difficult problems. We will begin with the simplest complex figure… two rectangles put together.

**Example 1****:** Find the area of the complex figure (two rectangles)

Solution:

One variation on this problem is when you aren’t given all of the lengths directly. In example 2, you must find a missing length before you are able to find the area of one of the rectangles.

**Example 2****:** Find the area of the complex figure (two rectangles)

Solution:

In example 2, a second way to do the problem is to split the figure up into a left rectangle and a right rectangle. A third way to do it is to visualize it as a large rectangle with the top left corner missing, then subtract the missing area from the total area of the rectangle. Regardless of how you do the problem, the answer is 150 in^{2}.

When one or more of the individual polygons is a triangle, trapezoid, or parallelogram, simply use the appropriate formula to find the area of each polygon, then add the areas to get a total. Examples 3 and 4 are complex figures that are comprised of individual polygons.

**Example 3****:** Find the area of the complex figure (rectangle and triangle)

Solution:

First, find the length and width of the rectangle and the base and height of the triangle. The rectangle’s length (16 m) and width (6 m) are given. The triangle has a given height (8 m) but the base is not given. It is represented by the length of 5 m and the blue dotted line segment. The length of the blue segment can be found by subtracting the length of the bottom of the rectangle from the length of the top left corner of the rectangle (16 m – 6m). The length of the blue segment is 10 m, and the entire base of the triangle is 15 m.

**Example 4****:** Find the area of the complex figure (two parallelograms)

Solution:

The total height is 12 meters. However, the problem doesn’t specify what the individual heights are for each parallelogram. It appears that they are the same size, so the height of each would be 6. However, since both figures are parallelograms whose bases are the same length, the actual division of the heights doesn’t change the overall area. If the heights are 7 and 5 or 8 and 4, the area would be the same.

A second way to do this problem would be to combine both parallelograms into the same formula and use the base of 10 and the total height of 12.