3.2 Use Properties of Rectangles, Triangles, and Trapezoids

Izabela Mazur

CHAPTER 3 Measurement, Perimeter, Area, and Volume

3.2 Use Properties of Rectangles, Triangles, and Trapezoids

Learning Objectives

By the end of this section, you will be able to:

Understand linear, square, and cubic measure
Use properties of rectangles
Use properties of triangles
Use properties of 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.1). A tape measure might remind you of a line—you use it for linear measure, which measures length. Inch, foot, yard, mile, centimetre and metre are units of linear measure.

This tape measure measures inches along the top and centimetres along the bottom.

A picture of a portion of a tape measure is shown. The top shows the numbers 1 through 5. The portion from the beginning to the 1 has a red circle and an arrow to a picture from 0 to 1 inch, with 1 sixteenth, 1 eighth, 3 eighths, 1 half, and 3 fourths labeled. Above this, it is labeled “Standard Measures.” The bottom of the tape measure shows the numbers 1 through 10, then 1 and 2. The region from the edge to about 3 and a half has a red circle with an arrow pointing to a picture from 0 to 3.5. It is labeled 0, 1 cm, 1.7 cm, 2.3 cm and 3.5 cm. Above this, it is labeled “Metric (S).” — Figure.1

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 centimetres, or square miles to measure area. A square centimetre is a square that is one centimetre (cm) on each side. A square inch is a square that is one inch on each side (Figure.2).

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

Two squares are shown. The smaller one has sides labeled 1 cm and is 1 square centimetre. The larger one has sides labeled 1 inch and is 1 square inch. — Figure.2

(Figure.3) 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.

A rectangle is shown. It has 3 squares across and 2 squares down, a total of 6 squares. — Figure 3 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 centimetres. When measuring the volume of a rectangular solid, you measure how many cubes fill the container. We often use cubic centimetres, cubic inches, and cubic feet. A cubic centimetre is a cube that measures one centimetre on each side, while a cubic inch is a cube that measures one inch on each side (Figure.4).

Two cubes are shown. The smaller one has sides labeled 1 cm and is labeled as 1 cubic centimetre. The larger one has sides labeled 1 inch and is labeled as 1 cubic inch. — Figure 4 Cubic measures have sides that are 1 unit in length.

Suppose the cube in (Figure.5) 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.

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

A cube is shown, comprised of smaller cubes. Each side of the cube has 3 smaller cubes across, for a total of 27 smaller cubes. — Figure.5

EXAMPLE 1

For each item, state whether you would use linear, square, or cubic measure:

a) amount of carpeting needed in a room

b) extension cord length

c) amount of sand in a sandbox

d) length of a curtain rod

e) amount of flour in a canister

f) size of the roof of a doghouse.

Solution

a) You are measuring how much surface the carpet covers, which is the area.	square measure
b) You are measuring how long the extension cord is, which is the length.	linear measure
c) You are measuring the volume of the sand.	cubic measure
d) You are measuring the length of the curtain rod.	linear measure
e) You are measuring the volume of the flour.	cubic measure
f) You are measuring the area of the roof.	square measure

TRY IT 1.1

Determine whether you would use linear, square, or cubic measure for each item.

a) amount of paint in a can b) height of a tree c) floor of your bedroom d) diametre of bike wheel e) size of a piece of sod f) amount of water in a swimming pool

Show answer

cubic
linear
square
linear
square
cubic

TRY IT 1.2

Determine whether you would use linear, square, or cubic measure for each item.

a) volume of a packing box b) size of patio c) amount of medicine in a syringe d) length of a piece of yarn e) size of housing lot f) height of a flagpole

Show answer

cubic
square
cubic
linear
square
linear

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.

Perimeter and Area

The perimeter is a measure of the distance around a figure.

The area is a measure of the surface covered by a figure.

(Figure. 6) 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.

$\begin{array}{c}\text{Perimeter}=4\phantom{\rule{0.2em}{0ex}}\text{inches}\\ \text{Area}=1\phantom{\rule{0.2em}{0ex}}\text{square inch}\end{array}$

A 5 square by 5 square checkerboard is shown with each side labeled 1 inch. An image of an ant is shown on the top left square. — Figure 6 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 2

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

a) What is the perimeter of the figure?

b) What is the area?

A checkerboard is shown. It has 10 squares across the top and 5 down the side.

Solution

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

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

A checkerboard is shown. It has 10 squares across the top and 5 down the side. The top and bottom each have two adjacent 1 inch labels across, the sides have 1 inch labels.

TRY IT 2.1

Find the a) perimeter and b) area of the figure:

A rectangle is shown comprised of 3 squares.

Show answer

8 inches
3 sq. inches

TRY IT 2.2

Find the a) perimeter and b) area of the figure:

A square is shown comprised of 4 smaller squares.

Show answer

8 centimetres
4 sq. centimetres

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.7).

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

A rectangle is shown. Each angle is marked with a square. The top and bottom are labeled L, the sides are labeled W. — Figure.7

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

$\begin{array}{c}P=L+W+L+W\hfill \\ \hfill \text{or}\hfill \\ P=2L+2W\hfill \end{array}$

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.8). 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 $A=L\cdot W$ .

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

A rectangle is shown. It is made up of 6 squares. The bottom is 2 squares across and marked as 2, the side is 3 squares long and marked as 3. — Figure.8

Properties of Rectangles

Rectangles have four sides and four right $\left(\text{90}\right)$ ° angles.
The lengths of opposite sides are equal.
The perimeter, $P$ , of a rectangle is the sum of twice the length and twice the width. See (Figure 8). $P=2L+2W$

The area, $A$ , of a rectangle is the length times the width. $A=L\cdot W$

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

HOW TO: Use a Problem Solving Strategy for Geometry Applications

Read the problem and make sure you understand all the words and ideas. Draw the figure and label it with the given information.
Identify what you are looking for.
Name what you are looking for. Choose a variable to represent that quantity.
Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.
Solve the equation using good algebra techniques.
Check the answer in the problem and make sure it makes sense.
Answer the question with a complete sentence.

EXAMPLE 3

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

Solution

a)
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 metres.

b)
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 metres.

TRY IT 3.1

The length of a rectangle is $120$ yards and the width is $50$ yards. Find a) the perimeter and b) the area.

Show answer

340 yd
6000 sq. yd

TRY IT 3.2

The length of a rectangle is $62$ feet and the width is $48$ feet. Find a) the perimeter and b) the area.

Show answer

220 ft
2976 sq. ft

EXAMPLE 4

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.

TRY IT 4.1

Find the length of a rectangle with a perimeter of $80$ inches and width of $25$ inches.

Show answer

15 in.

TRY IT 4.2

Find the length of a rectangle with a perimeter of $30$ yards and width of $6$ yards.

Show answer

9 yd

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 5

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.

TRY IT 5.1

The width of a rectangle is seven metres less than the length. The perimeter is $58$ metres. Find the length and width.

Show answer

18 m, 11 m

TRY IT 5.2

The length of a rectangle is eight feet more than the width. The perimeter is $60$ feet. Find the length and width.

Show answer

11 ft , 19 ft

EXAMPLE 6

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

TRY IT 6.1

The length of a rectangle is eight more than twice the width. The perimeter is $64$ feet. Find the length and width.

Show answer

8 ft, 24 ft

TRY IT 6.2

The width of a rectangle is six less than twice the length. The perimeter is $18$ centimetres. Find the length and width.

Show answer

5 cm, 4 cm

EXAMPLE 7

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.

TRY IT 7.1

The area of a rectangle is $598$ square feet. The length is $23$ feet. What is the width?

Show answer

26 ft

TRY IT 7.2

The width of a rectangle is $21$ metres. The area is $609$ square metres. What is the length?

Show answer

29 m

EXAMPLE 8

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.

TRY IT 8.1

The perimeter of a rectangular swimming pool is $200$ feet. The length is $40$ feet more than the width. Find the length and width.

Show answer

30 ft, 70 ft

TRY IT 8.2

The length of a rectangular garden is $30$ yards more than the width. The perimeter is $300$ yards. Find the length and width.

Show answer

60 yd, 90 yd

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), we’ve labeled the length $b$ and the width $h$ , so it’s area is $bh$ .

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

A rectangle is shown. The side is labeled h and the bottom is labeled b. The centre says A equals bh. — Figure.9

We can divide this rectangle into two congruent triangles (Figure.10). 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 $A=\frac{1}{2}bh$ .

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

A rectangle is shown. A diagonal line is drawn from the upper left corner to the bottom right corner. The side of the rectangle is labeled h and the bottom is labeled b. Each triangle says one-half bh. To the right of the rectangle, it says “Area of each triangle,” and shows the equation A equals one-half bh. — Figure.10

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.11) shows three triangles with the base and height of each marked.

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.

Three triangles are shown. The triangle on the left is a right triangle. The bottom is labeled b and the side is labeled h. The middle triangle is an acute triangle. The bottom is labeled b. There is a dotted line from the top vertex to the base of the triangle, forming a right angle with the base. That line is labeled h. The triangle on the right is an obtuse triangle. The bottom of the triangle is labeled b. The base has a dotted line extended out and forms a right angle with a dotted line to the top of the triangle. The vertical line is labeled h. — Figure.11

Triangle Properties

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

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

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, $h$ .

$A=\frac{1}{2}\phantom{\rule{0.1em}{0ex}}bh$

A triangle is shown. The vertices are labeled A, B, and C. The sides are labeled a, b, and c. There is a vertical dotted line from vertex B at the top of the triangle to the base of the triangle, meeting the base at a right angle. The dotted line is labeled h.

EXAMPLE 9

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.

TRY IT 9.1

Find the area of a triangle with base $13$ inches and height $2$ inches.

Show answer

13 sq. in.

TRY IT 9.2

Find the area of a triangle with base $14$ inches and height $7$ inches.

Show answer

49 sq. in.

EXAMPLE 10

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.

TRY IT 10.1

The perimeter of a triangular garden is $24$ feet. The lengths of two sides are $18$ feet and $22$ feet. How long is the third side?

Show answer

8 ft

TRY IT 10.2

The lengths of two sides of a triangular window are $7$ feet and $5$ feet. The perimeter is $18$ feet. How long is the third side?

Show answer

6 ft

EXAMPLE 11

The area of a triangular church window is $90$ square metres. The base of the window is $15$ metres. 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 metres.

TRY IT 11.1

The area of a triangular painting is $126$ square inches. The base is $18$ inches. What is the height?

Show answer

14 in.

TRY IT 11.2

A triangular tent door has an area of $15$ square feet. The height is $5$ feet. What is the base?

Show answer

6 ft

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.12) shows both types of triangles.

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.

Two triangles are shown. All three sides of the triangle on the left are labeled s. It is labeled “equilateral triangle”. Two sides of the triangle on the right are labeled s. It is labeled “isosceles triangle”. — Figure.12

Isosceles and Equilateral Triangles

An isosceles triangle has two sides the same length.

An equilateral triangle has three sides of equal length.

EXAMPLE 12

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 12.1

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

Show answer

13 in.

TRY IT 12.2

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

Show answer

17 cm

EXAMPLE 13

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 l

TRY IT 13.1

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?

Show answer

14 ft

TRY IT 13.2

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

Show answer

7 m

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.13).

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

A trapezoid is shown. The top is labeled b and marked as the smaller base. The bottom is labeled B and marked as the larger base. A vertical line forms a right angle with both bases and is marked as h. — Figure.13

Formula for the Area of a Trapezoid

${\text{Area}}_{\text{trapezoid}}=\frac{1}{2}h\left(b+B\right)$

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.14).

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

An image of a trapezoid is shown. The top is labeled with a small b, the bottom with a big B. A diagonal is drawn in from the upper left corner to the bottom right corner. — Figure.14

The height of the trapezoid is also the height of each of the two triangles. See (Figure.15).

The formula for the area of a trapezoid is

This image shows the formula for the area of a trapezoid and says “area of trapezoid equals one-half h times smaller base b plus larger base B).

If we distribute, we get,

The top line says area of trapezoid equals one-half times blue little b times h plus one-half times red big B times h. Below this is area of trapezoid equals A sub blue triangle plus A sub red triangle.

Properties of Trapezoids

A trapezoid has four sides. See (Figure.13).
Two of its sides are parallel and two sides are not.
The area, $A$ , of a trapezoid is $\text{A}=\frac{1}{2}h\left(b+B\right)$ .

EXAMPLE 14

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.

TRY IT 14.1

The height of a trapezoid is $14$ yards and the bases are $7$ and $16$ yards. What is the area?

Show answer

161 sq. yd

TRY IT 14.2

The height of a trapezoid is $18$ centimetres and the bases are $17$ and $8$ centimetres. What is the area?

Show answer

225 sq. cm

EXAMPLE 15

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.

TRY IT 15.1

The height of a trapezoid is $7$ centimetres and the bases are $4.6$ and $7.4$ centimetres. What is the area?

Show answer

42 sq. cm

TRY IT 15.2

The height of a trapezoid is $9$ metres and the bases are $6.2$ and $7.8$ metres. What is the area?

Show answer

63 sq. m

EXAMPLE 16

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 plan

TRY IT 16.1

Lin wants to sod his lawn, which is shaped like a trapezoid. The bases are $10.8$ yards and $6.7$ yards, and the height is $4.6$ yards. How many square yards of sod does he need?

Show answer

40.25 sq. yd

TRY IT 16.2

Kira wants cover his patio with concrete pavers. If the patio is shaped like a trapezoid whose bases are $18$ feet and $14$ feet and whose height is $15$ feet, how many square feet of pavers will he need?

Show answer

240 sq. ft.

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Key Concepts

Properties of Rectangles
- Rectangles have four sides and four right (90°) angles.
- The lengths of opposite sides are equal.
- The perimeter, , of a rectangle is the sum of twice the length and twice the width.
  - $P=2L+2W$
- The area, , of a rectangle is the length times the width.
  - $A=L\cdot W$
Triangle Properties
- For any triangle , the sum of the measures of the angles is 180°.
  - $m\angle A+m\angle B+m\angle C=180$ °
- 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.
  - $A=\frac{1}{2}bh$

Glossary

area: The area is a measure of the surface covered by a figure.

equilateral triangle: A triangle with all three sides of equal length is called an equilateral triangle.

isosceles triangle: A triangle with two sides of equal length is called an isosceles triangle.

perimeter: The perimeter is a measure of the distance around a figure.

rectangle: A rectangle is a geometric figure that has four sides and four right angles.

trapezoid: A trapezoid is four-sided figure, a quadrilateral, with two sides that are parallel and two sides that are not.

Practice Makes Perfect

Understand Linear, Square, and Cubic Measure

In the following exercises, determine whether you would measure each item using linear, square, or cubic units.

1. amount of water in a fish tank	2. length of dental floss
3. living area of an apartment	4. floor space of a bathroom tile
5. height of a doorway	6. capacity of a truck trailer

In the following exercises, find the a) perimeter and b) area of each figure. Assume each side of the square is $1$ cm.

7.	8.
9.	10.
11.	12.

Use the Properties of Rectangles

In the following exercises, find the a) perimeter and b) area of each rectangle.

13. The length of a rectangle is $85$ feet and the width is $45$ feet.	14. The length of a rectangle is $26$ inches and the width is $58$ inches.
15. A rectangular room is $15$ feet wide by $14$ feet long.	16. A driveway is in the shape of a rectangle $20$ feet wide by $35$ feet long.

In the following exercises, solve.

17. Find the length of a rectangle with perimeter $124$ inches and width $38$ inches.	18. Find the length of a rectangle with perimeter $20.2$ yards and width of $7.8$ yards.
19. Find the width of a rectangle with perimeter $92$ metres and length $19$ metres.	20. Find the width of a rectangle with perimeter $16.2$ metres and length $3.2$ metres.
21. The area of a rectangle is $414$ square metres. The length is $18$ metres. What is the width?	22. The area of a rectangle is $782$ square centimetres. The width is $17$ centimetres. What is the length?
23. The length of a rectangle is $9$ inches more than the width. The perimeter is $46$ inches. Find the length and the width.	24. The width of a rectangle is $8$ inches more than the length. The perimeter is $52$ inches. Find the length and the width.
25. The perimeter of a rectangle is $58$ metres. The width of the rectangle is $5$ metres less than the length. Find the length and the width of the rectangle.	26. The perimeter of a rectangle is $62$ feet. The width is $7$ feet less than the length. Find the length and the width.
27. The width of the rectangle is $0.7$ metres less than the length. The perimeter of a rectangle is $52.6$ metres. Find the dimensions of the rectangle.	28. The length of the rectangle is $1.1$ metres less than the width. The perimeter of a rectangle is $49.4$ metres. Find the dimensions of the rectangle.
29. The perimeter of a rectangle of $150$ feet. The length of the rectangle is twice the width. Find the length and width of the rectangle.	30. The length of a rectangle is three times the width. The perimeter is $72$ feet. Find the length and width of the rectangle.
31. The length of a rectangle is $3$ metres less than twice the width. The perimeter is $36$ metres. Find the length and width.	32. The length of a rectangle is $5$ inches more than twice the width. The perimeter is $34$ inches. Find the length and width.
33. The width of a rectangular window is $24$ inches. The area is $624$ square inches. What is the length?	34. The length of a rectangular poster is $28$ inches. The area is $1316$ square inches. What is the width?
35. The area of a rectangular roof is $2310$ square metres. The length is $42$ metres. What is the width?	36. The area of a rectangular tarp is $132$ square feet. The width is $12$ feet. What is the length?
37. The perimeter of a rectangular courtyard is $160$ feet. The length is $10$ feet more than the width. Find the length and the width.	38. The perimeter of a rectangular painting is $306$ centimetres. The length is $17$ centimetres more than the width. Find the length and the width.
39. The width of a rectangular window is $40$ inches less than the height. The perimeter of the doorway is $224$ inches. Find the length and the width.	40. The width of a rectangular playground is $7$ metres less than the length. The perimeter of the playground is $46$ metres. Find the length and the width.

Use the Properties of Triangles

In the following exercises, solve using the properties of triangles.

41. Find the area of a triangle with base $12$ inches and height $5$ inches.	42. Find the area of a triangle with base $45$ centimetres and height $30$ centimetres.
43. Find the area of a triangle with base $8.3$ metres and height $6.1$ metres.	44. Find the area of a triangle with base $24.2$ feet and height $20.5$ feet.
45. A triangular flag has base of $1$ foot and height of $1.5$ feet. What is its area?	46. A triangular window has base of $8$ feet and height of $6$ feet. What is its area?
47. If a triangle has sides of $6$ feet and $9$ feet and the perimeter is $23$ feet, how long is the third side?	48. If a triangle has sides of $14$ centimetres and $18$ centimetres and the perimeter is $49$ centimetres, how long is the third side?
49. What is the base of a triangle with an area of $207$ square inches and height of $18$ inches?	50. What is the height of a triangle with an area of $893$ square inches and base of $38$ inches?
51. The perimeter of a triangular reflecting pool is $36$ yards. The lengths of two sides are $10$ yards and $15$ yards. How long is the third side?	52. A triangular courtyard has perimeter of $120$ metres. The lengths of two sides are $30$ metres and $50$ metres. How long is the third side?
53. An isosceles triangle has a base of $20$ centimetres. If the perimeter is $76$ centimetres, find the length of each of the other sides.	54. An isosceles triangle has a base of $25$ inches. If the perimeter is $95$ inches, find the length of each of the other sides.
55. Find the length of each side of an equilateral triangle with a perimeter of $51$ yards.	56. Find the length of each side of an equilateral triangle with a perimeter of $54$ metres.
57. The perimeter of an equilateral triangle is $18$ metres. Find the length of each side.	58. The perimeter of an equilateral triangle is $42$ miles. Find the length of each side.
59. The perimeter of an isosceles triangle is $42$ feet. The length of the shortest side is $12$ feet. Find the length of the other two sides.	60. The perimeter of an isosceles triangle is $83$ inches. The length of the shortest side is $24$ inches. Find the length of the other two sides.
61. A dish is in the shape of an equilateral triangle. Each side is $8$ inches long. Find the perimeter.	62. A floor tile is in the shape of an equilateral triangle. Each side is $1.5$ feet long. Find the perimeter.
63. A road sign in the shape of an isosceles triangle has a base of $36$ inches. If the perimeter is $91$ inches, find the length of each of the other sides.	64. A scarf in the shape of an isosceles triangle has a base of $0.75$ metres. If the perimeter is $2$ metres, find the length of each of the other sides.
65. The perimeter of a triangle is $39$ feet. One side of the triangle is $1$ foot longer than the second side. The third side is $2$ feet longer than the second side. Find the length of each side.	66. The perimeter of a triangle is $35$ feet. One side of the triangle is $5$ feet longer than the second side. The third side is $3$ feet longer than the second side. Find the length of each side.
67. One side of a triangle is twice the smallest side. The third side is $5$ feet more than the shortest side. The perimeter is $17$ feet. Find the lengths of all three sides.	68. One side of a triangle is three times the smallest side. The third side is $3$ feet more than the shortest side. The perimeter is $13$ feet. Find the lengths of all three sides.

Use the Properties of Trapezoids

In the following exercises, solve using the properties of trapezoids.

69. The height of a trapezoid is $12$ feet and the bases are $9$ and $15$ feet. What is the area?	70. The height of a trapezoid is $24$ yards and the bases are $18$ and $30$ yards. What is the area?
71. Find the area of a trapezoid with a height of $51$ metres and bases of $43$ and $67$ metres.	72. Find the area of a trapezoid with a height of $62$ inches and bases of $58$ and $75$ inches.
73. The height of a trapezoid is $15$ centimetres and the bases are $12.5$ and $18.3$ centimetres. What is the area?	74. The height of a trapezoid is $48$ feet and the bases are $38.6$ and $60.2$ feet. What is the area?
75. Find the area of a trapezoid with a height of $4.2$ metres and bases of $8.1$ and $5.5$ metres.	76. Find the area of a trapezoid with a height of $32.5$ centimetres and bases of $54.6$ and $41.4$ centimetres.
77. Laurel is making a banner shaped like a trapezoid. The height of the banner is $3$ feet and the bases are $4$ and $5$ feet. What is the area of the banner?	78. Niko wants to tile the floor of his bathroom. The floor is shaped like a trapezoid with width $5$ feet and lengths $5$ feet and $8$ feet. What is the area of the floor?
79. Theresa needs a new top for her kitchen counter. The counter is shaped like a trapezoid with width $18.5$ inches and lengths $62$ and $50$ inches. What is the area of the counter?	80. Elena is knitting a scarf. The scarf will be shaped like a trapezoid with width $8$ inches and lengths $48.2$ inches and $56.2$ inches. What is the area of the scarf?

Everyday Math

81. Fence Jose just removed the children’s playset from his back yard to make room for a rectangular garden. He wants to put a fence around the garden to keep out the dog. He has a $50$ foot roll of fence in his garage that he plans to use. To fit in the backyard, the width of the garden must be $10$ feet. How long can he make the other side if he wants to use the entire roll of fence?

82. Gardening Lupita wants to fence in her tomato garden. The garden is rectangular and the length is twice the width. It will take $48$ feet of fencing to enclose the garden. Find the length and width of her garden.

83. Fence Christa wants to put a fence around her triangular flowerbed. The sides of the flowerbed are $6$ feet, $8$ feet, and $10$ feet. The fence costs $\$10$ per foot. How much will it cost for Christa to fence in her flowerbed?

84. Painting Caleb wants to paint one wall of his attic. The wall is shaped like a trapezoid with height $8$ feet and bases $20$ feet and $12$ feet. The cost of the painting one square foot of wall is about $?0.05$ . About how much will it cost for Caleb to paint the attic wall?

A right trapezoid is shown.

Writing Exercises

86. If you need to put a fence around your backyard, do you need to know the perimeter or the area of the backyard? Explain your reasoning.

87. Look at the two figures.

A rectangle is shown on the left. It is labeled as 2 by 8. A square is shown on the right. It is labeled as 4 by 4.

a) Which figure looks like it has the larger area? Which looks like it has the larger perimeter?

b) Now calculate the area and perimeter of each figure. Which has the larger area? Which has the larger perimeter?

88. The length of a rectangle is $5$ feet more than the width. The area is $50$ square feet. Find the length and the width.

a) Write the equation you would use to solve the problem.

b) Why can’t you solve this equation with the methods you learned in the previous chapter?

Answers

1. cubic	3. square	5. linear
7. a) 10 cm b) 4 sq. cm	9. a) 8 cm b) 3 sq. cm	11. a) 10 cm b) 5 sq. cm
13. a) 260 ft b) 3825 sq. ft	15. a) 58 ft b) 210 sq. ft	17. 24 inches
19. 27 metres	21. 23 m	23. 7 in., 16 in.
25. 17 m, 12 m	27. 13.5 m, 12.8 m	29. 25 ft, 50 ft
31. 7 m, 11 m	33. 26 in.	35. 55 m
37. 35 ft, 45 ft	39. 76 in., 36 in.	41. 60 sq. in.
43. 25.315 sq. m	45. 0.75 sq. ft	47. 8 ft
49. 23 in.	51. 11 ft	53. 28 cm
55. 17 ft	57. 6 m	59. 15 ft
61. 24 in.	63. 27.5 in.	65. 12 ft, 13 ft, 14 ft
67. 3 ft, 6 ft, 8 ft	69. 144 sq. ft	71. 2805 sq. m
73. 231 sq. cm	75. 28.56 sq. m	77. 13.5 sq. ft
79. 1036 sq. in.	81. 15 ft	83. $24
85. Answers will vary.	87. Answers will vary.

Attributions

This chapter has been adapted from “Use Properties of Rectangles, Triangles, and Trapezoids” in Prealgebra (OpenStax) by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, which is under a CC BY 4.0 Licence. Adapted by Izabela Mazur. See the Copyright page for more information.

License

Icon for the Creative Commons Attribution 4.0 International License

Understand Linear, Square, and Cubic Measure

Use the Properties of Rectangles

Use the Properties of Triangles

Isosceles and Equilateral Triangles

Use the Properties of Trapezoids

Key Concepts

Glossary

Practice Makes Perfect

Understand Linear, Square, and Cubic Measure

Use the Properties of Rectangles

Use the Properties of Triangles

Use the Properties of Trapezoids

Everyday Math

Writing Exercises

Answers

Attributions

License

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