9.4 Use Properties of Rectangles, Triangles, and Trapezoids

Understand Linear, Square, and Cubic Measure

When you measure your height or the length of a garden hose, you use a ruler or tape measure (Figure 9.13). A tape measure might remind you of a line—you use it for linear measure, which measures length. Inch, foot, yard, mile, centimeter and meter are units of linear measure.

Figure 9.13 This tape measure measures inches along the top and centimeters along the bottom.

When you want to know how much tile is needed to cover a floor, or the size of a wall to be painted, you need to know the area, a measure of the region needed to cover a surface. Area is measured is square units. We often use square inches, square feet, square centimeters, or square miles to measure area. A square centimeter is a square that is one centimeter (cm) on each side. A square inch is a square that is one inch on each side (Figure 9.14).

Figure 9.14 Square measures have sides that are each $1$ unit in length.

Figure 9.15 shows a rectangular rug that is $2$ feet long by $3$ feet wide. Each square is $1$ foot wide by $1$ foot long, or $1$ square foot. The rug is made of $6$ squares. The area of the rug is$6$ square feet.

Figure 9.15 The rug contains six squares of 1 square foot each, so the total area of the rug is 6 square feet.

When you measure how much it takes to fill a container, such as the amount of gasoline that can fit in a tank, or the amount of medicine in a syringe, you are measuring volume. Volume is measured in cubic units such as cubic inches or cubic centimeters. When measuring the volume of a rectangular solid, you measure how many cubes fill the container. We often use cubic centimeters, cubic inches, and cubic feet. A cubic centimeter is a cube that measures one centimeter on each side, while a cubic inch is a cube that measures one inch on each side (Figure 9.16).

Figure 9.16 Cubic measures have sides that are 1 unit in length.

Suppose the cube in Figure 9.17 measures $3$ inches on each side and is cut on the lines shown. How many little cubes does it contain? If we were to take the big cube apart, we would find $27$ little cubes, with each one measuring one inch on all sides. So each little cube has a volume of $1$ cubic inch, and the volume of the big cube is $27$ cubic inches.

Figure 9.17 A cube that measures 3 inches on each side is made up of 27 one-inch cubes, or 27 cubic inches.

Many geometry applications will involve finding the perimeter or the area of a figure. There are also many applications of perimeter and area in everyday life, so it is important to make sure you understand what they each mean.

Picture a room that needs new floor tiles. The tiles come in squares that are a foot on each side—one square foot. How many of those squares are needed to cover the floor? This is the area of the floor.

Next, think about putting new baseboard around the room, once the tiles have been laid. To figure out how many strips are needed, you must know the distance around the room. You would use a tape measure to measure the number of feet around the room. This distance is the perimeter.

Figure 9.18 shows a square tile that is $1$ inch on each side. If an ant walked around the edge of the tile, it would walk $4$ inches. This distance is the perimeter of the tile.

Since the tile is a square that is $1$ inch on each side, its area is one square inch. The area of a shape is measured by determining how many square units cover the shape.

Figure 9.18 $\begin{array}{c}\text{Perimeter}=4\text{inches}\\ \text{Area}=1\text{square inch}\end{array}$
When the ant walks completely around the tile on its edge, it is tracing the perimeter of the tile. The area of the tile is 1 square inch.

Example 9.26

Each of two square tiles is $1$ square inch. Two tiles are shown together.

1. ⓐ What is the perimeter of the figure?

2. ⓑ What is the area?

Solution

ⓐ The perimeter is the distance around the figure. The perimeter is $6$ inches.

ⓑ The area is the surface covered by the figure. There are $2$ square inch tiles so the area is $2$ square inches.

Try It 9.51

Find the ⓐ perimeter and ⓑ area of the figure:

Try It 9.52

Find the ⓐ perimeter and ⓑ area of the figure:

Use the Properties of Rectangles

A rectangle has four sides and four right angles. The opposite sides of a rectangle are the same length. We refer to one side of the rectangle as the length, $L,$ and the adjacent side as the width, $W.$ See Figure 9.19.

Figure 9.19 A rectangle has four sides, and four right angles. The sides are labeled L for length and W for width.

The perimeter, $P,$ of the rectangle is the distance around the rectangle. If you started at one corner and walked around the rectangle, you would walk $L+W+L+W$ units, or two lengths and two widths. The perimeter then is

What about the area of a rectangle? Remember the rectangular rug from the beginning of this section. It was $2$ feet long by $3$ feet wide, and its area was $6$ square feet. See Figure 9.20. Since $A=2\cdot 3,$ we see that the area, $A,$ is the length, $L,$ times the width, $W,$ so the area of a rectangle is $A=L\cdot W.$

Figure 9.20 The area of this rectangular rug is $6$ square feet, its length times its width.

For easy reference as we work the examples in this section, we will restate the Problem Solving Strategy for Geometry Applications here.

Example 9.27

The length of a rectangle is $32$ meters and the width is $20$ meters. Find ⓐ the perimeter, and ⓑ the area.

Solution

ⓐ
Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	the perimeter of a rectangle
Step 3. Name. Choose a variable to represent it.	Let P = the perimeter
Step 4. Translate. Write the appropriate formula. Substitute.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.	The perimeter of the rectangle is 104 meters.

ⓑ
Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	the area of a rectangle
Step 3. Name. Choose a variable to represent it.	Let A = the area
Step 4. Translate. Write the appropriate formula. Substitute.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.	The area of the rectangle is 60 square meters.

Example 9.28

Find the length of a rectangle with perimeter $50$ inches and width $10$ inches.

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	the length of the rectangle
Step 3. Name. Choose a variable to represent it.	Let L = the length
Step 4. Translate. Write the appropriate formula. Substitute.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.	The length is 15 inches.

In the next example, the width is defined in terms of the length. We’ll wait to draw the figure until we write an expression for the width so that we can label one side with that expression.

Example 9.29

The width of a rectangle is two inches less than the length. The perimeter is $52$ inches. Find the length and width.

Solution

Step 1. Read the problem.
Step 2. Identify what you are looking for.	the length and width of the rectangle
Step 3. Name. Choose a variable to represent it. Now we can draw a figure using these expressions for the length and width.	Since the width is defined in terms of the length, we let L = length. The width is two feet less that the length, so we let L − 2 = width
Step 4.Translate. Write the appropriate formula. The formula for the perimeter of a rectangle relates all the information. Substitute in the given information.
Step 5. Solve the equation.	$52=2L+2L-4$
Combine like terms.	$52=4L-4$
Add 4 to each side.	$56=4L$
Divide by 4.	$\frac{56}{4}=\frac{4L}{4}$
	$14=L$
	The length is 14 inches.
Now we need to find the width.
The width is L − 2.	The width is 12 inches.
Step 6. Check: Since $14+12+14+12=52$, this works!
Step 7. Answer the question.	The length is 14 feet and the width is 12 feet.

Example 9.30

The length of a rectangle is four centimeters more than twice the width. The perimeter is $32$ centimeters. Find the length and width.

Solution

Step 1. Read the problem.
Step 2. Identify what you are looking for.	the length and width
Step 3. Name. Choose a variable to represent it.	let W = width The length is four more than twice the width. 2w + 4 = length
Step 4.Translate. Write the appropriate formula and substitute in the given information.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.	The length is 12 cm and the width is 4 cm.

Example 9.31

The area of a rectangular room is $168$ square feet. The length is $14$ feet. What is the width?

Solution

Step 1. Read the problem.
Step 2. Identify what you are looking for.	the width of a rectangular room
Step 3. Name. Choose a variable to represent it.	Let W = width
Step 4.Translate. Write the appropriate formula and substitute in the given information.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.	The width of the room is 12 feet.

Example 9.32

The perimeter of a rectangular swimming pool is $150$ feet. The length is $15$ feet more than the width. Find the length and width.

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	the length and width of the pool
Step 3. Name. Choose a variable to represent it. The length is 15 feet more than the width.	Let $W=\text{width}$ $W+15=\text{length}$
Step 4.Translate. Write the appropriate formula and substitute.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.	The length of the pool is 45 feet and the width is 30 feet.

Use the Properties of Triangles

We now know how to find the area of a rectangle. We can use this fact to help us visualize the formula for the area of a triangle. In the rectangle in Figure 9.20, we’ve labeled the length $b$ and the width $h,$ so it’s area is $bh.$

Figure 9.21 The area of a rectangle is the base, $b,$ times the height, $h.$

We can divide this rectangle into two congruent triangles (Figure 9.22). Triangles that are congruent have identical side lengths and angles, and so their areas are equal. The area of each triangle is one-half the area of the rectangle, or $\frac{1}{2}bh.$ This example helps us see why the formula for the area of a triangle is $A=\frac{1}{2}bh.$

Figure 9.22 A rectangle can be divided into two triangles of equal area. The area of each triangle is one-half the area of the rectangle.

The formula for the area of a triangle is $A=\frac{1}{2}bh,$ where $b$ is the base and $h$ is the height.

To find the area of the triangle, you need to know its base and height. The base is the length of one side of the triangle, usually the side at the bottom. The height is the length of the line that connects the base to the opposite vertex, and makes a $\text{90°}$ angle with the base. Figure 9.23 shows three triangles with the base and height of each marked.

Figure 9.23 The height $h$ of a triangle is the length of a line segment that connects the the base to the opposite vertex and makes a $\text{90°}$ angle with the base.

Triangle Properties

For any triangle $\text{Δ}ABC,$ the sum of the measures of the angles is $\text{180°}.$

$$m\text{∠}A+m\text{∠}B+m\text{∠}C=\text{180°}$$

The perimeter of a triangle is the sum of the lengths of the sides.

$$P=a+b+c$$

The area of a triangle is one-half the base, $b,$ times the height, $h.$

$$A=\frac{1}{2}bh$$

Example 9.33

Find the area of a triangle whose base is $11$ inches and whose height is $8$ inches.

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	the area of the triangle
Step 3. Name. Choose a variable to represent it.	let A = area of the triangle
Step 4.Translate. Write the appropriate formula. Substitute.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.	The area is 44 square inches.

Example 9.34

The perimeter of a triangular garden is $24$ feet. The lengths of two sides are $4$ feet and $9$ feet. How long is the third side?

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	length of the third side of a triangle
Step 3. Name. Choose a variable to represent it.	Let c = the third side
Step 4.Translate. Write the appropriate formula. Substitute in the given information.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.	The third side is 11 feet long.

Example 9.35

The area of a triangular church window is $90$ square meters. The base of the window is $15$ meters. What is the window’s height?

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	height of a triangle
Step 3. Name. Choose a variable to represent it.	Let h = the height
Step 4.Translate. Write the appropriate formula. Substitute in the given information.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.	The height of the triangle is 12 meters.

Isosceles and Equilateral Triangles

Besides the right triangle, some other triangles have special names. A triangle with two sides of equal length is called an isosceles triangle. A triangle that has three sides of equal length is called an equilateral triangle. Figure 9.24 shows both types of triangles.

Figure 9.24 In an isosceles triangle, two sides have the same length, and the third side is the base. In an equilateral triangle, all three sides have the same length.

Isosceles and Equilateral Triangles

An isosceles triangle has two sides the same length.

An equilateral triangle has three sides of equal length.

Example 9.36

The perimeter of an equilateral triangle is $93$ inches. Find the length of each side.

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.	Perimeter = 93 in.
Step 2. Identify what you are looking for.	length of the sides of an equilateral triangle
Step 3. Name. Choose a variable to represent it.	Let s = length of each side
Step 4.Translate. Write the appropriate formula. Substitute.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.	Each side is 31 inches.

Try It 9.71

Find the length of each side of an equilateral triangle with perimeter $39$ inches.

Try It 9.72

Find the length of each side of an equilateral triangle with perimeter $51$ centimeters.

Example 9.37

Arianna has $156$ inches of beading to use as trim around a scarf. The scarf will be an isosceles triangle with a base of
$60$ inches. How long can she make the two equal sides?

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.	P = 156 in.
Step 2. Identify what you are looking for.	the lengths of the two equal sides
Step 3. Name. Choose a variable to represent it.	Let s = the length of each side
Step 4.Translate. Write the appropriate formula. Substitute in the given information.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.	Arianna can make each of the two equal sides 48 inches long.

Try It 9.73

A backyard deck is in the shape of an isosceles triangle with a base of $20$ feet. The perimeter of the deck is $48$ feet. How long is each of the equal sides of the deck?

Try It 9.74

A boat’s sail is an isosceles triangle with base of $8$ meters. The perimeter is $22$ meters. How long is each of the equal sides of the sail?

Use the Properties of Trapezoids

A trapezoid is four-sided figure, a quadrilateral, with two sides that are parallel and two sides that are not. The parallel sides are called the bases. We call the length of the smaller base $b,$ and the length of the bigger base $B.$ The height, $h,$ of a trapezoid is the distance between the two bases as shown in Figure 9.25.

Figure 9.25 A trapezoid has a larger base, $B,$ and a smaller base, $b.$ The height $h$ is the distance between the bases.

The formula for the area of a trapezoid is:

Splitting the trapezoid into two triangles may help us understand the formula. The area of the trapezoid is the sum of the areas of the two triangles. See Figure 9.26.

Figure 9.26 Splitting a trapezoid into two triangles may help you understand the formula for its area.

The height of the trapezoid is also the height of each of the two triangles. See Figure 9.27.

Figure 9.27

The formula for the area of a trapezoid is

If we distribute, we get,

Example 9.38

Find the area of a trapezoid whose height is 6 inches and whose bases are $14$ and $11$ inches.

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	the area of the trapezoid
Step 3. Name. Choose a variable to represent it.	Let $A=\text{the area}$
Step 4.Translate. Write the appropriate formula. Substitute.
Step 5. Solve the equation.
Step 6. Check: Is this answer reasonable?

If we draw a rectangle around the trapezoid that has the same big base $B$ and a height $h,$ its area should be greater than that of the trapezoid.

If we draw a rectangle inside the trapezoid that has the same little base $b$ and a height $h,$ its area should be smaller than that of the trapezoid.

The area of the larger rectangle is $84$ square inches and the area of the smaller rectangle is $66$ square inches. So it makes sense that the area of the trapezoid is between $84$ and $66$ square inches

Step 7. Answer the question. The area of the trapezoid is $75$ square inches.

Example 9.39

Find the area of a trapezoid whose height is $5$ feet and whose bases are $10.3$ and $13.7$ feet.

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	the area of the trapezoid
Step 3. Name. Choose a variable to represent it.	Let A = the area
Step 4.Translate. Write the appropriate formula. Substitute.
Step 5. Solve the equation.
Step 6. Check: Is this answer reasonable? The area of the trapezoid should be less than the area of a rectangle with base 13.7 and height 5, but more than the area of a rectangle with base 10.3 and height 5.
Step 7. Answer the question.	The area of the trapezoid is 60 square feet.

Example 9.40

Vinny has a garden that is shaped like a trapezoid. The trapezoid has a height of $3.4$ yards and the bases are $8.2$ and $5.6$ yards. How many square yards will be available to plant?

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	the area of a trapezoid
Step 3. Name. Choose a variable to represent it.	Let A = the area
Step 4.Translate. Write the appropriate formula. Substitute.
Step 5. Solve the equation.
Step 6. Check: Is this answer reasonable? Yes. The area of the trapezoid is less than the area of a rectangle with a base of 8.2 yd and height 3.4 yd, but more than the area of a rectangle with base 5.6 yd and height 3.4 yd.
Step 7. Answer the question.	Vinny has 23.46 square yards in which he can plant.