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1.2 Use Properties of Rectangles, Triangles and Quadrilaterals

Learning Objectives

  • 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, centimeter and meter are units of linear measure.
The top of a tape measure shows the numbers 1 through 5 in inches. The bottom of the tape measure shows the numbers 1 through 10, then 1 and 2 in cemtimeters.
Figure 1 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 2).
Two squares are shown. The smaller one has sides labeled 1 cm and is 1 square centimeter. The larger one has sides labeled 1 inch and is 1 square inch.
Figure 2 Square measures have sides that are each 1 unit in length.
Figure 3 below 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 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 4).
Two cubes are shown. The smaller one has sides 1 cm and is 1 cubic centimeter. The larger one has sides 1 inch and is 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 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 A cube that measures 3 inches on each side is made up of 27 one-inch cubes, or 27 cubic inches.

Choosing Linear, Square, or Cubic Measure

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.

 

Show Solution to (a)
You are measuring how much surface the carpet covers, which is the area. Square Measure
Show Solution to (b)
You are measuring how long the extension cord is, which is the length. Linear Measure
Show Solution to (c)
You are measuring the volume of the sand. Cubic Measure
Show Solution to (d)
You are measuring the length of the curtain rod. Linear Measure
Show Solution to (e)
You are measuring the volume of the flour. Cubic Measure
Show Solution to (f)
You are measuring the surface area of the roof. Square Measure

Try It 

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) diameter of bike wheel

(e) size of a piece of sod

(f) amount of water in a swimming pool

Show Solution

(a) Cubic

(b) Linear

(c) Square

(d) Linear

(e) Square

(f) Cubic

Try It

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 Solution

(a) Cubic

(b) Square

(c) Cubic

(d) Linear

(e) Square

(f) 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.

A 5x5 chess board.
Figure 6 Perimeter = 4 inches; Area = 1 square inch 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.

Finding Perimeter and Area of a Rectangle

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

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

(a) What is the perimeter of the figure?

(b) What is the area?

Show Solution to (a)

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

A checkerboard with 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.

Show Solution to (b)

(b) The area is the surface covered by the figure. There are 2 square inch tiles. Therefore, the area is 2 square inches.

A checkerboard with 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  

Each box in the figure below is 1 square inch. Find the (a) perimeter and (b) area of the figure:


A rectangle is shown comprised of 3 squares.

Show Solution

Perimeter = 8 inches
Area = 3 square inches

Try It  

Each box in the figure below is 1 square inch. Find the (a) perimeter and (b) area of the figure:


A square is shown comprised of 4 smaller squares.

Show Solution

Perimeter = 8 inches
Area = 4 square inches

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

P=L+W+L+WorP=2L+2W

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⋅3, we see that the area, A, is the length, L, times the width, W, so the area of a rectangle is A = L⋅W.
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 The area of this rectangular rug is 6 square feet, its length times its width.

Properties of Rectangles

  • Rectangles have four sides and four right (90[latex]^\circ[/latex]) 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 7.
P=2L+2WP=2L+2W
  • The area, A, of a rectangle is the length times the width.
A=L⋅W

How To

Use a Problem Solving Strategy for Geometry Applications

Step 1. Read the problem and make sure you understand all the words and ideas. Draw the figure and label it with the given information.

Step 2. Identify what you are looking for.

Step 3. Name what you are looking for. Choose a variable to represent that quantity.

Step 4. Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.

Step 5. Solve the equation using good algebra techniques.

Step 6. Check the answer in the problem and make sure it makes sense.

Step 7. Answer the question with a complete sentence.

Finding Perimeter and Area of a Rectangle

The length of a rectangle is 32 meters and its width is 20 meters.  Find its (a) perimeter and (b) area.

Show Solution to (a)
Explanation Steps
Step 1. Read the problem. Draw the figure and label it with the given information. A rectangle with 32 m length and 20 m width.
 

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.

  

sub in 32 for L and 20 for W
 

Step 5. Solve the equation.

  

P=64+40=104
 

Step 6. Check:

 Sub in the values to get a true equation.
Step 7. Answer the question. The perimeter of the rectangle is 104 meters.
Show Solution to (b)
Explanation Steps
Step 1. Read the problem. Draw the figure and label it with the given information. A rectangle with 32 m length and 20 m width.
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 of the rectangle.
Step 4. Translate.
Write the appropriate formula.
Substitute.		 Sub in 32m for L and 20m for W.
Step 5. Solve the equation. A=640
Step 6. Check:

 

 Check by subbing in the values and getting a true statement.
Step 7. Answer the question. The area of the rectangle is 640 square meters.

Try It

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

Show Solution

Perimeter = 340 yards
Area = 6000 square yards

 

Try It 

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

Show Solution

Perimeter = 220 feet
Area = 2976 square feet

Finding Length of a Rectangle Given Perimeter

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

Show Solution
Explanation Steps
Step 1. Read the problem. Draw the figure and label it with the given information. Rectangle with length L and width 10 in.
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 of the rectangle
Step 4. Translate.

Write the appropriate formula.

Substitute.

 Sub in 50 for P and 10 for W.
 

Step 5. Solve the equation.

  

Steps to solve the equation.
Step 6. Check:

 

 Sub in 15 to check the problem and get a true statement.

 
Step 7. Answer the question. The length is 15 inches.

Try It

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

Show Solution

Length = 15 inches

Try It

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

Show Solution

Length = 9 yards

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.

Finding Length and Width of a Rectangle Given Perimeter

The width of a rectangle is two inches less than the length. The perimeter is 52 inches. Find the length and width.
Show Solution
Explanation Steps
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.

Rectangle with opposite sides labled L-2 and the remaining opposite sides labeled L.
 

Step 4.Translate.
Write the appropriate formula.The formula for the perimeter of a rectangle relates all the information.Substitute in the given information.

 Substitute into the P=2L+2W formula: 52=2L+2(L-2).
 

Step 5. Solve the equation.

  

52=2L+2L−452=2L+2L−4
Combine like terms. 52=4L−452=4L−4
Add 4 to each side. 56=4L56=4L
Divide by 4. 564=4L4564=4L4
14=L14=L
The length is 14 inches.
Now we need to find the width. The width is L − 2.  

 

[latex]L-2[/latex]

[latex]14-2 = 12[/latex]

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 inches and the width is 12 inches.

Try It

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

Show Solution

Length = 18 meters
Width = 11 meters

Try It

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

Show Solution

Length = 19 feet
Width = 11 feet

Finding Length and Width of a Rectangle Given Perimeter

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

Show Solution
Steps Explanation
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

Rectangle with opposite sides labled "w" and the remaining opposite sides labeled 2w+4.
 

Step 4.Translate.
Write the appropriate formula and substitute in the given information.

  

Substitute into the P=2L+2W formula: 32=2(2w+4)+2w.
Step 5. Solve the equation. [latex]32=4w+8+2w[/latex]

[latex]32=6w+8[/latex]

[latex]24=6w[/latex]

[latex]4=w[/latex] is the width

find the length: [latex]2w+4[/latex]

[latex]2(4)+4 = 12[/latex] so 12 is the length
Step 6. Check:

The check shows 32=32, so the solution is correct.
Step 7. Answer the question. The length is 12 cm and the width is 4 cm.

Try It 

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

Show Solution

Width = 8 feet
Length = 24 feet

Try It 

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

Show Solution

Width = 4 centimeters
Length = 5 centimeters

Finding Width of a Rectangle Given Area

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

Show Solution
Steps Explanation
 Step 1. Read the problem. Rectangle with one side labled W and the adjacent side of 14 ft. The Area = 186 square feet.
Step 2. Identify what you are looking for.  

the width of a rectangular room

 
Step 3. Name. Choose a variable to represent it. The length is 15 feet more than the width.  

 

Let W = width

 

 
Step 4.Translate. Write the appropriate formula and substitute in the given information. 168=14W

 
Step 5. Solve the equation. 12=W

 
Step 6. Check:

The check shows 168=168, so the solution is correct.

  

 

 
Step 7. Answer the question. The width of the room is 12 feet.

Try It

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

Show Solution

Width = 26 feet

Try It 

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

Show Solution

Length = 29 meters

Finding Length and Width of a Rectangle Given Perimeter

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

Show Solution
Steps Explanation
Step 1. Read the problem. Draw the figure and label it with the given information. Rectangle with one side labeled W and the adjacent sided labeled W+15. The Perimeter is 150 feet.
 

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 = width

W + 15 = length

 
Step 4.Translate.
Write the appropriate formula and substitute.		 Substitute into the P=2L+2W formula: 150=2(w+15)+2w.
Step 5. Solve the equation. [latex]150=2w+30+2w[/latex]

[latex]150=4w+30[/latex]

[latex]120=4w[/latex]

[latex]30=w[/latex] is the width of the pool

find the length: [latex]w+15[/latex]

[latex]30+15 = 45[/latex] is the length of the pool
Step 6. Check:

150=150, so the check is true.

  

 

 
Step 7. Answer the question.  

The length of the pool is 45 feet and the width is 30 feet.

Try It 

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 Solution

Length = 70 feet
Width = 30 feet

Try It 

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

Show Solution

Length = 90 yards
Width = 60 yards

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 its area is bh.
Rectangle with one side labeled h and the adjacent sided labeled b. A=bh.
Figure 9 The area of a rectangle is the base, b, times the height, h.
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 [latex]\frac{1}{2}bh[/latex].

This example helps us see why the formula for the area of a triangle is [latex]A = \frac{1}{2}bh[/latex] , where b is the base and h is the height.
A diagonal is drawn between the corners of a rectangle with sides h and b. Each triangle says one-half bh. Area of each triangle= one-half bh.
Figure 10 A rectangle can be divided into two triangles of equal area. The area of each triangle is one-half the area of the rectangle.
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 90° angle with the base. Figure 9.23 shows three triangles with the base and height of each marked.
Three triangles—right, acute, obtuse—each with base b and height h shown using dotted perpendiculars when needed.
Figure 11 The height h of a triangle is the length of a line segment that connects the base to the opposite vertex and makes a 90° angle with the base.

Triangle Properties

For any triangle ΔABC, the sum of the measures of the angles is 180°.
m∠A+m∠B+m∠C=180°m∠A+m∠B+m∠C=180°

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

P=a+b+cP=a+b+c

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

A=12bhA=12bhTriangle with vertices A, B, C; sides a, b, c; a dotted height h drops from B to the base at a right angle.

Finding Area of a Triangle

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

Show Solution
Steps Explanation
Step 1. Read the problem. Draw the figure and label it with the given information. Triangle with a dotted height of 8 inches and base of 11 inches.
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.

  

Substitute into the A=(1/2)*b*h formula: A=(1/2)*11*8.
 

Step 5. Solve the equation.

  

 [latex]A=44[/latex] square inches
Step 6. Check:
44=44 shows the check is true.
Step 7. Answer the question. The area is 44 square inches.

Finding Missing Side Length of a Triangle Given Perimeter

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?

Show Solution
Steps Explanation
Step 1. Read the problem. Draw the figure and label it with the given information. Triangle with sides c, 4 ft, and 9 ft. P=24 ft.
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.		 Substitute into the P=a+b+c formula: 24=4+9+c.
Step 5. Solve the equation. [latex]24=13+c[/latex]

[latex]11=c[/latex]
Step 6. Check:
24=24 shows the check is true.
Step 7. Answer the question. The third side is 11 feet long.

Try It

The perimeter of a triangular garden is 48 feet. The lengths of two sides are 18 feet and 22 feet. How long is the third side?

Show Solution

Third side = 8 feet

Try It

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 Solution

Third side = 6 feet

Finding Height of a Triangle Given Area

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?

Show Solution
Steps Explanation
Step 1. Read the problem. Draw the figure and label it with the given information. Isosceles triangle with base 15 m and height h.
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.		 Substitute into the A=(1/2)*b*h formula: 90=(1/2)*15*h.
 

Step 5. Solve the equation.

  

 

 [latex]90=\frac{15}{2}h[/latex]

[latex]12=h[/latex]
 

Step 6. Check:
90=90 shows the check is true.
Step 7. Answer the question. The height of the triangle is 12 meters.

Try It 

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

Show Solution

Height = 14 inches

Try It 

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

Show Solution

Base = 6 feet

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.
Two triangles are shown. Two sides of the isosceles triangle on the left are s. All three sides of the equilateral triangle on the left are labeled s.
Figure 12 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.

Finding Side Lengths of an Equilateral Triangle Given Perimeter

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

Show Solution
Steps Explanation
Step 1. Read the problem. Draw the figure and label it with the given information. All three sides of the equilateral triangle on the left are labeled s.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.

  

Substitute into the P=a+b+c formula: 93=s+s+s.
 

Step 5. Solve the equation.

  

 [latex]93=3s[/latex]

[latex]31=s[/latex]
 

Step 6. Check:
All three sides of the equilateral triangle on the left are labeled 31.
93=93 shows the check is true.
 

Step 7. Answer the question.

  

Each side is 31 inches.

Try It 

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

Show Solution

13 inches

Try It 

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

Show Solution

17 centimeters

Finding Side Lengths of an Isosceles Triangle

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?

Show Solution
Steps Explanation
Step 1. Read the problem. Draw the figure and label it with the given information. Two sides of an isosceles triangle are "s" and the base is 60 in.        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.

  

Substitute into the P=a+b+c formula: 156=s+60+s.
Step 5. Solve the equation.  

[latex]156=2s+60[/latex]

[latex]96=2s[/latex]

[latex]48=s[/latex]
Step 6. Check:
156=156 shows the check is true.
Step 7. Answer the question. Arianna can make each of the two equal sides 48 inches long.

Try It

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 Solution

14 feet

Try It

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?

Show Solution

7 meters

Use the Properties of Trapezoids

A trapezoid is a 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.
Trapezoid with small top base b and larger base B; a vertical height h forms right angles with both bases.
Figure 13 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:
Areatrapezoid=12h(b+B)Areatrapezoid=12h(b+B)

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.
A trapezoid with top side b, and bottom side big B. A diagonal is drawn in from the upper left corner to the bottom right corner.
Figure 14 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 15.
A diagonal splits a trapezoid, illustrating how the triangle can be rearranged to fit the shape of a rectangle.
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,


Area of trapezoid equals 1/2 * blue little b * h plus 1/2 * red big B * h. Below this substitutes blue triangle for b and red triangle for B.

Properties of Trapezoids

  • A trapezoid has four sides. See Figure 13.
  • Two of its sides are parallel and two sides are not.
  • The area is
    • A=12h(b+B)A=12h(b+B) 

Finding Area of a Trapezoid

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

Show Solution
Steps Explanation
Step 1. Read the problem. Draw the figure and label it with the given information. Trapezoid with B=14 in, b=11in, and h=6in.
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 areaA=the area
 

Step 4.Translate.
Write the appropriate formula.
Substitute.

  

Substitute into the A=(1/2)*b*(b+B) formula: A=(1/2)*6*(11+14).
Step 5. Solve the equation. [latex]A=\frac{1}{2} \cdot 6(25)[/latex]

[latex]A=3(25)[/latex]

Area is 75 square inches
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.

Chart shows trapezoid as rectangle (A=84), trapezoid alone (A=75), and trapezoid with inner rectangle (A=66).

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

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

Show Solution

Area = 225 square centimeters

Finding Area of a Trapezoid

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

Show Solution
Steps Explanation
Step 1. Read the problem. Draw the figure and label it with the given information.
Trapezoid with b=10.3 ft, B=13.7 ft, and h=5 ft.
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.

  

Substitute into the A=(1/2)*b*(b+B) formula: A=(1/2)*5*(10.3+13.7).
 

Step 5. Solve the equation.

  

[latex]A= \frac{1}{2} \cdot 5(24)[/latex]

[latex]A=12(5)[/latex]

Area is 60 square feet.
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.

Trapezoid with outer red rectangle and another with inner black rectangle; areas compare as 68.5 > 60 > 51.5.
Step 7. Answer the question. The area of the trapezoid is 60 square feet.

Try It

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

Show Solution

Area = 42 square centimeters

Try It

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

Show Solution

Area = 63 square meters

Finding Area of a Trapezoid

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?

Show Solution
Steps Explanation
Step 1. Read the problem. Draw the figure and label it with the given information. Trapezoid with b=5.6 yd, B=8.2 yd, and h=3.4 yd.
 

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.
Substitute into the A=(1/2)*b*(b+B) formula: A=(1/2)*3.4*(5.6+8.2).
 

Step 5. Solve the equation.

  

 

[latex]A= \frac{1}{2} (3.4)(13.8)[/latex]

Area is 23.46 square yards.
 

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.

Chart compares area work: rectangle 27.88, trapezoid 23.46, rectangle 19.04; shows formulas and values in each column.
 

Step 7. Answer the question.

  

Vinny has 23.46 square yards in which he can plant.

Try It

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 Solution

Area = 40.25 square yards

Try It

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 Solution

Area = 240 square feet

Key Concepts

Properties of Rectangles

  • Rectangles have four sides and four right (90°) 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: [latex]P = 2L + 2W[/latex]
  • The area, A, of a rectangle is the length times the width: [latex]A = LW[/latex]

Triangle Properties

  • For any triangle ΔABC, the sum of the measures of the angels is 180°: [latex]m\angle A + m\angle B + m\angle C = 180^\circ[/latex]
  • The perimeter of a triangle is the sum of the lengths of the sides: [latex]P = a + b + c[/latex]
  • The area of a triangle is one-half the base, b, times the height, h:  [latex]A = \frac{1}{2}bh[/latex]

 

Section Exercises

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
Show Solution
Cubic

2. length of dental floss

3. living area of an apartment

Show Solution
Square

4. floor space of a bathroom tile

5. height of a doorway

Show Solution
Linear

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. A rectangle is shown comprised of 4 squares forming a horizontal line


A rectangle is shown comprised of 4 squares forming a horizontal line.
Show Solution to (a)
10 cm
Show Solution to (b)
4 cm2

 

8. A rectangle is shown comprised of 3 squares forming a vertical line.

A rectangle is shown comprised of 3 squares forming a vertical line.

9. Three squares are shown. There is one on the bottom left, one on the bottom right, and one on the top right.


Three squares are shown. There is one on the bottom left, one on the bottom right, and one on the top right.
Show Solution to (a)
8 cm
Show Solution to (b)
3 cm2

10. Four squares are shown. Three form a horizontal line, and there is one above the center square.


Four squares are shown. Three form a horizontal line, and there is one above the center square.

11. There are three squares forming a horizontal line across the top and two underneath the two on the right.


Five squares are shown. There are three forming a horizontal line across the top and two underneath the two on the right.
Show Solution to (a)
10 cm
Show Solution to (b)
5 cm2 

12. A square is shown. It is comprised of nine smaller squares.


A square is shown. It is comprised of nine smaller squares.

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.

Show Solution to (a)
260 ft. 
Show Solution to (b)
3825 ft2

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.

Show Solution to (a)
58 ft. 
Show Solution to (b)
210 ft2

16. A driveway is in the shape of a rectangle 20 feet wide by 35 feet long.

 

17. Find the length of a rectangle with perimeter 124 inches and width 38 inches.

Show Solution
24 in. 

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 meters and length 19 meters. 

Show Solution
27 meters  

20. Find the width of a rectangle with perimeter 16.2 meters and length 3.2 meters

21. The area of a rectangle is 414 square meters. The length is 18 meters. What is the width?

Show Solution
23 meters

22. The area of a rectangle is 782 square centimeters. The width is 17 centimeters. 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.

Show Solution
7 in., 16 in.
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 meters. The width of the rectangle is 5 meters less than the length. Find the length and the width of the rectangle.

Show Solution
17 m, 12 m

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 meters less than the length. The perimeter of a rectangle is 52.6 meters. Find the dimensions of the rectangle.

Show Solution
13.5 m, 12.8 m

28. The length of the rectangle is 1.1 meters less than the width. The perimeter of a rectangle is 49.4 meters. 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.

Show Solution
25 ft, 50 ft

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 meters less than twice the width. The perimeter is 36 meters. Find the length and width.
Show Solution
7 m, 11 m

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?

Show Solution
26 in. 

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 meters. The length is 42 meters. What is the width?
Show Solution
55 m
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.

Show Solution
35 ft., 45 ft.

38. The perimeter of a rectangular painting is 306 centimeters. The length is 17 centimeters 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.

Show Solution
76 in., 36 in.

40. The width of a rectangular playground is 7 meters less than the length. The perimeter of the playground is 46 meters. Find the length and the width.

41. Find the area of a triangle with base 12 inches and height 5 inches.

Show Solution
30 sq. in.

42. Find the area of a triangle with base 45 centimeters and height 30 centimeters.

43. Find the area of a triangle with base 8.3 meters and height 6.1 meters.

Show Solution
25.315 sq. m

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?

Show Solution
0.75 sq. ft

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?

Show Solution
8 ft

48. If a triangle has sides of 14 centimeters and 18 centimeters and the perimeter is 49 centimeters, 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?

Show Solution

Solution

23 in. 

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?
Show Solution

Solution

11 yd

52. A triangular courtyard has perimeter of 120 meters. The lengths of two sides are 30 meters and 50 meters. How long is the third side?

53. An isosceles triangle has a base of 20 centimeters. If the perimeter is 76 centimeters, find the length of each of the other sides.

Show Solution

Solution

28 cm

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.

Show Solution

Solution

17 yd

56. Find the length of each side of an equilateral triangle with a perimeter of 54 meters.

57. The perimeter of an equilateral triangle is 18 meters. Find the length of each side.

Show Solution

Solution

6 m

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.

Show Solution

Solution

15 ft

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.

Show Solution

Solution

24 in.

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.

Show Solution

Solution

27.5 in. 

64. A scarf in the shape of an isosceles triangle has a base of 0.75 meters. If the perimeter is 2 meters, 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.

Show Solution

Solution

12 ft, 13 ft, 14 ft

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.

Show Solution

Solution

3 ft, 6 ft, 8 ft

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?

Show Solution

Solution

144 sq. ft

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 meters and bases of 43 and 67 meters.
Show Solution

Solution

2805 sq. m

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 centimeters and the bases are 12.5 and 18.3 centimeters. What is the area?
Show Solution

Solution

231 sq. cm

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 meters and bases of 8.1 and 5.5 meters.

Show Solution

Solution

28.56 sq. m

76. Find the area of a trapezoid with a height of 32.5 centimeters and bases of 54.6 and 41.4 centimeters.

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?

Show Solution

Solution

13.5 sq. ft

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?

Show Solution

Solution

1036 sq. in. 

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?

Show Solution

Solution

15 ft

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?

Show Solution

Solution

$240

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

85. If you need to put tile on your kitchen floor, do you need to know the perimeter or the area of the kitchen? Explain your reasoning.

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) 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?

A 2 by 8 rectangle and a 4 by 4 square.

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?

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.

triangle

A triangle is a geometric figure with three sides and three angles.

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