1. Module 4

1.11. Page 6

Mathematics 10-3 Module 4 Lesson 2

Module 4: Area

 

Bringing Ideas Together

 

In Explore you discovered that the area of a trapezoid is composed of the areas of two triangles.

 

For the trapezoid in Explore, with height h and parallel sides a and b,

 

 

 

 

One way of remembering this formula is to think of as the average of the lengths of the two parallel sides, a and b. So,

area of a trapezoid = average of the two parallel sides × height

 

The multimedia piece “Area of a Trapezoid” shows another way to explain why the formula works.

 

Work through the following examples to test the use of the average length of parallel sides to calculate area.

 

Example 4

 

A piece of fabric for a quilt is pictured below.

 

 

 

This illustration shows a trapezoid with height 4 inches and bases 4 inches and 10 inches.

 

What is the area of this piece of fabric?

 

Solution

 

Method 1

 

Apply the formula .

 

 

 

 

a = 10 in

b = 4 in

h = 4 in

 

 

 

 

The area of the piece of fabric is 28 in2.

 

Method 2

 

Apply the following formula.

 

 

 

area = average of the two parallel sides × height

 

To find the average of the parallel side, add the sides together and divide by 2.

 

 

 

 

So,

 

 

 

 

The area of the piece of fabric is 28 in2.

 

Method 3

 

Divide the trapezoid into two triangles by dividing from the upper-left vertex to the lower-right vertex. This shows that there are two triangles making up the trapezoid. Each triangle has a height of 4 in.

 

Lower Triangle

 

b = 4 in, h = 4 in

 

Upper Triangle

 

b = 10 in, h = 4 in

 

 

The area of the composite shape is the sum of the areas of these triangles.

 

 

 

 

The area of the piece of fabric is 28 in2.

 

More Composite Figures

 

Investigate how simple shapes can be combined to form composite figures in “Exploring Composite Figures.” (You may ignore circles for now; circles will be covered in the next lesson.)

 

In the next few examples you will examine areas of composite figures.

 

Example 5

 

As part of a community historical project, Anita is planning to paint a mural on the back of her garage.

 

 

 

This illustration shows the back of a garage with a total height of 12 ft and including a rectangular wall measuring 24 feet by 8 feet.

 

What area can Anita paint?

 

Solution

 

Step 1: Separate the composite figure into simpler shapes.

 

 

 

This illustration shows the back of a garage with the triangular and rectangular parts separated.

 

Step 2: Transfer the dimensions of the composite figure to these simpler shapes.

 

 

 

This illustration shows the back of a garage with a height of 12 feet split between the wall and the triangular end. The rectangular part of the wall measures 24 feet by 8 feet. The triangular part of the wall has a height of 4 feet and a base of 24 feet.

 

Step 3: Calculate the area of each simple shape.

 

 

 

 

 

 

 

Step 4: Combine the areas of the simpler shapes.

 

 

 

 

Anita could paint a mural 240 ft2 in area on the back of her garage.

 

Now it’s your turn!

 

Self-Check

 

SC 7. The following flag is used by maritime ships to signal “starboard.”

 

 

 

This diagram shows a maritime signal flag. It is a starboard pennant, which is the shape of a trapezoid with a height of 3 feet and bases of 8 inches and 1 foot 4 inches.

 

What is the area of this flag?

 

SC 8. Hong is going to apply some fertilizer to the lawn next to her house. The lawn is shaped like this.

 

 

 

This illustration shows a composite shape with a doghouse cutout. The shape consists of two rectangles. One is 9 metres wide by 12 metres long. Both rectangles fit into a larger rectangle measuring 20 metres by 12 metres. The second rectangle’s width is 9 metres less than the larger rectangle; its length is 7 metres less than the larger rectangle. The doghouse measures 1.5 metres by 1 metre.

 

What is the area of the lawn?

 

SC 9. A stop sign is a regular octagon. Each side is 10 in long. The distance from the top of the sign to the bottom is approximately 24.14 in.

 

 

 

This illustration shows an octagonal stop sign with each side of the octagon measuring 10 inches. The total height of the stop sign is about 24.14 inches.

 

What is the area of the stop sign to the nearest square inch?

 

SC 10. Look at the composite shape.

 

 

 

This illustration shows a composite shape. The total height is 15 millimetres. The top measurement is 30 millimetres. The bottom measure is 40 millimetres. There is a rectangular cutout 20 millimetres deep with a height one-third of the total height of the shape.

 

You can infer that the top and bottom of the shape are parallel from the small squares in the left corners of the illustration. There is a rectangular cutout 20 mm deep with a height one-third of the total height of the shape.

 

Calculate the area of the composite shape.

 

Compare your answers.

 

Mastering Concepts

 

Try this question. When you are finished, check your answer.

 

In Explore, you derived the formula for a trapezoid by dividing the shape into two triangles. You could have found the formula by dividing the trapezoid into a parallelogram and a triangle.

 

This illustration shows a trapezoid with parallel sides of length a and b. The height is h.

 

Show that this method also gives you the formula .

 

Compare your answers.