Area of a Complex Figure: Trapezoid

Lesson Plan - Get It!

Audio:

Complex figures are made up of more than one shape.

Sometimes, complex figures are called irregular shapes because they are a combination of regular shapes. Irregular shapes do not have all equal sides or equal angles.

There is not one formula you can use to find area, and you have to break the irregular shape into regular shapes that you know the area formula for such as circles, squares, rectangles, and triangles.

Before you learn how to decompose, or break apart, a complex figure, review the area formulas for regular shapes:

First, look at the irregular shape below:

To find the area of this shape, we need to decompose this complex shape into smaller regular shapes.

We can find the areas of the regular shapes and add them together to solve for the area of this larger irregular shape.

• Do you see any regular shapes?

Two rectangles make up this complex shape. We know the formula and how to find the area of rectangles!

Now, we just need to find the measurements for these rectangles. Then, we will use the formula to find the area of the red rectangle::

Next, find the measurements and then the area for the blue rectangle:

Finally, to find the area of the larger complex shape, simply add the two areas together:

The area of this complex shape is 137 square inches!

Next, let's look at a trapezoid.

A trapezoid can also be considered a complex, or irregular, shape:

• What regular shapes can a trapezoid be decomposed into?

A trapezoid can be decomposed into a rectangle and two identical right triangles.

Now, let's find the measurements for each regular shape.

To find the base measurements for each triangle, we need to start with the long base of the trapezoid and subtract the short length of the trapezoid:

10 - 3 = 7

Since both right triangles are identical, their bases will be the same. Divide the remaining trapezoid base length in half to find the triangle bases:

7 ÷ 2 = 3.5

Now, use the formulas we know to find the area of each regular shape:

Last, add each area together to find the total area for the trapezoid:

7 + 12 + 7 = 26

The total area of this trapezoid is 26 cm².

Although a trapezoid can be considered a complex shape because it can be decomposed into smaller regular shapes, it has a formula to find the area.

Area of a trapezoid = ½ (b1 + b2) (h)

Look at the same trapezoid again. This time use the formula to find the area.

The same answer! Both methods work.

Remember not all complex shapes have formulas. Many, like our very first example, need to be decomposed into smaller regular shapes like squares, rectangles, or triangles.

Watch the video below to learn more about the formula used to find the area of a trapezoid. As you watch, respond to the following in your math journal:

1. What is the formula for a trapezoid?
2. What does each part of the formula represent?
   • A = _______
   • b₁ = _______
   • b₂ = _______
   • h = _______
3. Why might it be helpful to label the parts of the trapezoid according to the formula parts?

Area of a trapezoid from Math Meeting:

In your math journal, write a response to the question:

• Would you rather decompose a complex figure and find the area of smaller shapes or use one formula to find the area of the entire shape? Explain.

In the Got It? section, you will practice calculating the area of complex figures in interactive games and practice.