7.1: Use Properties of Rectangles, Triangles, & Trapezoids
Chapter 7: Learning Outcomes
By the end of this chapter, you should be able to:
- Find perimeters of triangles, squares, rectangles, parallelograms, trapezoids, circles and composite figures using formulas.
- Find areas of the above shapes using formulas.
- Find the surface area of cubes, rectangular solids, right cylinders and cones, spheres, and composite solids using formulas.
- Distinguish between concepts of perimeter and area and their respective units.
Understand Linear, Square, & Cubic Measure
When you measure your height or the length of a garden hose, you use a ruler or tape measure (see Figure 7.1.1). 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.
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 in 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 (see Figure 7.1.2).
Square measures have sides that are each 1 unit in length.
Figure 7.1.3 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.
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 (see Figure 7.1.4).
Suppose the cube in Figure 7.1.5 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, 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.
Example A
For each item, state whether you would use linear, square, or cubic measure:
- amount of carpeting needed in a room
- extension cord length
- amount of sand in a sandbox
- length of a curtain rod
- amount of flour in a canister
- size of the roof of a doghouse
- You are measuring how much surface the carpet covers, which is the area. Square measure
- You are measuring how long the extension cord is, which is the length. Linear measure
- You are measuring the volume of the sand. Cubic measure
- You are measuring the length of the curtain rod. Linear measure
- You are measuring the volume of the flour. Cubic measure
- You are measuring the area of the roof. Square measure
Exercise 1
Determine whether you would use linear, square, or cubic measure for each item.
- amount of paint in a can
- height of a tree
- floor of your bedroom
- diametre of bike wheel
- size of a piece of sod
- amount of water in a swimming pool
Exercise 1 Answers
- cubic
- linear
- square
- linear
- square
- cubic
Exercise 2
Determine whether you would use linear, square, or cubic measure for each item.
- volume of a packing box
- size of patio
- amount of medicine in a syringe
- length of a piece of yarn
- size of housing lot
- height of a flagpole
Exercise 2 Answers
- cubic
- square
- cubic
- linear
- square
- linear
Many geometry applications will involve finding the perimeter or 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 a 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 7.1.6 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.
Perimeter = 4 inches
Area = 1 square inch
Example B
Each of the two square tiles is 1 square inch. Two tiles are shown together (see Figure 7.1.7).
a. Perimeter
The perimeter is the distance around the figure. The perimeter is 6 inches (see Figure 7.1.8).
b. Area
The area is the surface covered by the figure. There are 2 square inch tiles so the area is 2 square inches.
Exercise 3
For Figure 7.1.9, find the a) perimeter and b) area.
Exercise 3 Answers
- 8 inches
- 3 sq. inches
Exercise 4
For Figure 7.1.10, find the a) perimeter and b) area.
Exercise 4 Answers
- 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 7.1.11).
A rectangle has four sides and four right angles. The sides are labelled 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 [latex]L+W+L+W[/latex] units or two lengths and two widths. The perimeter, then, is
[latex]\begin{array}{c}P=L+W+L+W\hfill \\ \hfill \text{or}\hfill \\ P=2L+2W\hfill \end{array}[/latex]
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 7.1.12). Since [latex]A=2\cdot 3[/latex], we see that the area, A, is the length, L, times the width, W, so the area of a rectangle is [latex]A=L\cdot W[/latex].
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 [latex]\left(\text{90}\right)[/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 7.1.12). [latex]P=2L+2W[/latex]
- The area, A, of a rectangle is the length times the width. [latex]A=L\cdot W[/latex]
For easy reference, as we work through the examples in this section, we will state the problem-solving strategy for geometry applications here.
The problem-solving strategy for geometry applications is:
- 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 to the problem and make sure it makes sense.
- Answer the question with a complete sentence.
Example C
The length of a rectangle is 32 metres, and the width is 20 metres. Find the a) perimeter and b) area.
a. Perimeter
Step 1: Read the problem. Draw the figure and label it with the given information (see Figure 7.1.13).
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.
[latex]\begin{equation}\begin{split}P&=2L+2W \\ P&=2(32)+2(20) \end{split}\end{equation}[/latex]
Step 5: Solve the equation.
[latex]\begin{equation}\begin{split}P&=64+40 \\&=104\end{split}\end{equation}[/latex]
Step 6: Check.
[latex]\begin{equation}\begin{split}P &= 104? \\ 20+32+20+32 &= 104? \\ 104 &= 104\text{ ✔} \\ \end{split}\end{equation}[/latex]
Step 7: Answer the question.
The perimeter of the rectangle is 104 metres.
b. Area
Step 1: Read the problem. Draw the figure and label it with the given information (same as in part a).
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.
[latex]\begin{equation}\begin{split}A&=L\cdot W \\ A&=32\text{ m}\cdot 20\text{ m}\end{split}\end{equation}[/latex]
Step 5: Solve the equation.
[latex]A = 640[/latex]
Step 6: Check.
[latex]\begin{equation}\begin{split}A &= 640? \\ 32\cdot20 &= 640? \\ 640 &= 640\text{ ✔} \\ \end{split}\end{equation}[/latex]
Step 7: Answer the question.
The area of the rectangle is 60 square metres.
Exercise 5
The length of a rectangle is 120 yards, and the width is 50 yards. Find the a) perimeter and b) area.
Exercise 5 Answers
- 340 yd
- 6000 sq. yd
Exercise 6
The length of a rectangle is 62 feet, and the width is 48 feet. Find the a) perimeter and b) area.
Exercise 6 Answers
- 220 ft
- 2976 sq. ft
Example D
Find the length of a rectangle with a perimeter of 50 inches and a width of 10 inches.
Step 1: Read the problem. Draw the figure and label it with the given information (see Figure 7.1.14).
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.
[latex]\begin{equation}\begin{split}P&=2L+2W \\ 50&=2L+2(10) \end{split}\end{equation}[/latex]
Step 5: Solve the equation.
[latex]\begin{equation}\begin{split}50-20&=2L+20-20 \\ 30&=2L \\ \dfrac{30}{2} &= \dfrac{2L}{2} \\ 15&=L \end{split}\end{equation}[/latex]
Step 6: Check.
[latex]\begin{equation}\begin{split}P &= 50 \\ 15+10+15+10 &= 50? \\ 50 &= 50\text{ ✔} \\ \end{split}\end{equation}[/latex]
Step 7: Answer the question.
The length is 15 inches.
Exercise 7
Find the length of a rectangle with a perimeter of 80 inches and a width of 25 inches.
Exercise 7 Answer
15 in.
Exercise 8
Find the length of a rectangle with a perimeter of 30 yards and a width of 6 yards.
Exercise 8 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 E
The width of a rectangle is two inches less than the length. The perimeter is 52 inches. Find the length and width.
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.
Since the width is defined in terms of the length, we let L = length. The width is two feet less than the length, so we let L − 2 = width
Now, we can draw a figure using these expressions for the length and width (see Figure 7.1.15).
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.
- Combine like terms.
- Add 4 to each side.
- Divide by 4.
[latex]\begin{equation}\begin{split} 52&=2L+2L-4 \\ 52&=4L-4 \\ 56&=4L \\ \dfrac{56}{4}&=\dfrac{4L}{4} \\ 14&=L \end{split}\end{equation}[/latex]
The length is 14 inches.
Now, we need to find the width. The width is [latex]L - 2[/latex].
[latex]\begin{array}{c} L-2 \\ 14-2 \\ 12\end{array}[/latex]
The width is 12 inches.
Step 6: Check.
Since [latex]14+12+14+12=52[/latex], this works!
Step 7: Answer the question.
The length is 14 feet, and the width is 12 feet.
Exercise 9
The width of a rectangle is seven metres less than the length. The perimeter is 58 metres. Find the length and width.
Exercise 9 Answers
18 m, 11 m
Exercise 10
The length of a rectangle is eight feet more than the width. The perimeter is 60 feet. Find the length and width.
Exercise 10 Answers
11 ft, 19 ft
Example F
The length of a rectangle is four centimetres more than twice the width. The perimeter is 32 centimetres. Find the length and width.
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.
[latex]2w + 4 = \text{ length}[/latex]
Now, you can draw your figure (see Figure 7.1.16).
Step 4: Translate. Write the appropriate formula and substitute the given information into it.
[latex]\begin{equation}\begin{split}P&=2L+2W \\ 32&=2(2w+4)+2w \end{split}\end{equation}[/latex]
Step 5: Solve the equation.
[latex]\begin{equation}\begin{split}32&=4w+8+2w \\ 32&=6w+8 \\ 24&=6w \\ 4&=w \end{split}\end{equation}[/latex]
The width is 4.
[latex]\begin{equation}\begin{split} 2w&+4\\ 2(4)&+4 \\ 12& \end{split}\end{equation}[/latex]
The length is 12.
Step 6: Check.
[latex]\begin{equation}\begin{split}P&=2L+2W \\ 32 &= 2\cdot 12 + 2\cdot 4 ? \\ 32&=32 \text{ ✔} \end{split}\end{equation}[/latex]
Step 7: Answer the question.
The length is 12 cm and the width is 4 cm.
Exercise 11
The length of a rectangle is eight more than twice the width. The perimeter is 64 feet. Find the length and width.
Exercise 11 Answers
8 ft, 24 ft
Exercise 12
The width of a rectangle is six less than twice the length. The perimeter is 18 centimetres. Find the length and width.
Exercise 12 Answers
5 cm, 4 cm
Example G
The area of a rectangular room is 168 square feet. The length is 14 feet. What is the width?
Step 1: Read the problem. Draw the figure and label it with the given information. (see Figure 7.1.17).
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.
[latex]\begin{equation}\begin{split}A&=LW \\ 168&=14W \end{split}\end{equation}[/latex]
Step 5: Solve the equation.
[latex]\begin{equation}\begin{split}\dfrac{168}{14}&=\dfrac{14W}{14} \\ 12&=W \end{split}\end{equation}[/latex]
Step 6: Check.
[latex]\begin{equation}\begin{split}A&=LW \\ 168&=14\cdot 12 ? \\ 168&=168 \text{ ✔} \end{split}\end{equation}[/latex]
Step 7: Answer the question.
The width of the room is 12 feet.
Exercise 13
The area of a rectangle is 598 square feet. The length is 23 feet. What is the width?
Exercise 13 Answers
26 ft
Exercise 14
The width of a rectangle is 21 metres. The area is 609 square metres. What is the length?
Exercise 14 Answers
29 m
Example H
The perimeter of a rectangular swimming pool is 150 feet. The length is 15 feet more than the width. Find the length and width.
Step 1: Read the problem. Draw the figure and label it with the given information (see Figure 7.1.18).
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
[latex]W+15=\text{ length}[/latex]
Step 4: Translate. Write the appropriate formula and substitute.
[latex]\begin{equation}\begin{split}P&=2L+2W \\ 150&=2(w+15)+2w\end{split}\end{equation}[/latex]
Step 5: Solve the equation.
[latex]\begin{equation}\begin{split}150&=2w+30+2w \\ 150&=4w+30 \\ 120&=4w \\ 30&=w\end{split}\end{equation}[/latex]
The width is 30.
[latex]\begin{equation}\begin{split}w&+15 \\ 30&+15 \\ 45& \end{split}\end{equation}[/latex]
The length is 45
Step 6: Check.
[latex]\begin{equation}\begin{split}P&=2L+2W \\ 150 &= 2(45) + 2(30) ? \\ 150&=150 \text{ ✔} \end{split}\end{equation}[/latex]
Step 7: Answer the question.
The length of the pool is 45 feet, and the width is 30 feet.
Exercise 15
The perimeter of a rectangular swimming pool is 200 feet. The length is 40 feet more than the width. Find the length and width.
Exercise 15 Answers
30 ft, 70 ft
Exercise 16
The length of a rectangular garden is 30 yards more than the width. The perimeter is 300 yards. Find the length and width.
Exercise 16 Answers
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 7.1.19, we’ve labelled the length b and the width h, so its area is bh.
The area of a rectangle is the base, b, times the height, h.
We can divide this rectangle into two congruent triangles (see Figure 7.1.20). 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]\tfrac{1}{2}bh[/latex]. This example helps us see why the formula for the area of a triangle is [latex]A=\tfrac{1}{2}bh[/latex].
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 [latex]A=\tfrac{1}{2}bh[/latex], 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 90° angle with the base. Figure 7.1.21 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 base to the opposite vertex and makes a 90° angle with the base.
Properties of triangles:
For any triangle [latex]\delta ABC[/latex], the sum of the measures of the angles is 180°.
[latex]m\angle A+m\angle B+m\angle C=\text{180}°[/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=\dfrac{1}{2}\phantom{\rule{0.1em}{0ex}}bh[/latex]
Figure 7.1.22 shows an example triangle.
Example I
Find the area of a triangle whose base is 11 inches and whose height is 8 inches.
Step 1: Read the problem. Draw the figure and label it with the given information (see Figure 7.1.23).
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.
[latex]\begin{equation}\begin{split}A&=\dfrac{1}{2} \cdot b \cdot h \\ A &= \dfrac{1}{2} \cdot 11 \cdot 8\end{split}\end{equation}[/latex]
Step 5: Solve the equation.
[latex]A=44\text{ square inches}[/latex]
Step 6: Check.
[latex]\begin{equation}\begin{split}A&=\dfrac{1}{2}bh \\ 44 &= \dfrac{1}{2}(11)(8) ? \\ 44&=44 \text{ ✔}\end{split}\end{equation}[/latex]
Step 7: Answer the question.
The area is 44 square inches.
Exercise 17
Find the area of a triangle with a base of 13 inches and a height of 2 inches.
Exercise 17 Answer
13 sq. in.
Exercise 18
Find the area of a triangle with a base of 14 inches and a height of 7 inches.
Exercise 18 Answer
49 sq. in.
Example J
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?
Step 1: Read the problem. Draw the figure and label it with the given information (see Figure 7.1.24).
Step 2: Identify what you are looking for.
The 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.
[latex]\begin{equation}\begin{split}P&=a+b+c \\ 24&=4+9+c \end{split}\end{equation}[/latex]
Step 5: Solve the equation.
[latex]\begin{equation}\begin{split}24&=13+c \\ 11&=c\end{split}\end{equation}[/latex]
Step 6: Check.
[latex]\begin{equation}\begin{split}P&=a+b+c \\ 24&=4+9+11 \\ 24&=24 \text{ ✔} \end{split}\end{equation}[/latex]
Step 7: Answer the question.
The third side is 11 feet long.
Exercise 19
The perimeter of a triangular garden is 24 feet. The lengths of the two sides are 18 feet and 22 feet. How long is the third side?
Exercise 19 Answer
8 ft
Exercise 20
The lengths of the two sides of a triangular window are 7 feet and 5 feet. The perimeter is 18 feet. How long is the third side?
Exercise 20 Answer
6 ft
Example K
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?
Step 1: Read the problem. Draw the figure and label it with the given information (see Figure 7.1.25).
Step 2: Identify what you are looking for.
The 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.
[latex]\begin{equation}\begin{split}A&=\dfrac{1}{2} \cdot b \cdot h \\ 90&=\dfrac{1}{2} \cdot 15 \cdot h \end{split}\end{equation}[/latex]
Step 5: Solve the equation.
[latex]\begin{equation}\begin{split} 90&=\dfrac{15}{2}h \\ 12&=h \end{split}\end{equation}[/latex]
Step 6: Check.
[latex]\begin{equation}\begin{split} A&=\dfrac{1}{2}bh \\ 90&=\dfrac{1}{2} \cdot 15 \cdot 12 ? \\ 90&=90 \text{ ✔} \end{split}\end{equation}[/latex]
Step 7: Answer the question.
The height of the triangle is 12 metres.
Exercise 21
The area of a triangular painting is 126 square inches. The base is 18 inches. What is the height?
Exercise 21 Answer
14 in.
Exercise 22
A triangular tent door has an area of 15 square feet. The height is 5 feet. What is the base?
Exercise 22 Answer
6 ft
Isosceles & 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 7.1.26 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.
An isosceles triangle has two sides the same length.
An equilateral triangle has three sides of equal length.
Example L
The perimeter of an equilateral triangle is 93 inches. Find the length of each side.
Step 1: Read the problem. Draw the figure and label it with the given information (see Figure 7.1.27).
Perimeter = 93 in.
Step 2: Identify what you are looking for.
The 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.
[latex]\begin{equation}\begin{split}P&=a+b+c \\ 93&=s+s+s \end{split}\end{equation}[/latex]
Step 5: Solve the equation.
[latex]\begin{equation}\begin{split}93&=3s \\31&=s \end{split}\end{equation}[/latex]
Step 6: Check.
Figure 7.1.28 shows the completed figure.
[latex]\begin{equation}\begin{split} 93&=31+31+31 ?\\ 93&=93\text{ ✔} \end{split}\end{equation}[/latex]
Step 7: Answer the question.
Each side is 31 inches.
Exercise 23
Find the length of each side of an equilateral triangle with a perimeter of 39 inches.
Exercise 23 Answer
13 in.
Exercise 24
Find the length of each side of an equilateral triangle with a perimeter of 51 centimetres.
Exercise 24 Answer
17 cm
Example M
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?
Step 1: Read the problem. Draw the figure and label it with the given information (see Figure 7.1.29).
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.
[latex]\begin{equation}\begin{split}P&=a+b+c \\ 156&=s+60+s \end{split}\end{equation}[/latex]
Step 5: Solve the equation.
[latex]\begin{equation}\begin{split} 156&=2s+60 \\ 96&=2s \\ 48&=s \end{split}\end{equation}[/latex]
Step 6: Check.
[latex]\begin{equation}\begin{split} 156&=48+60+48 ?\\ 156&=156\text{ ✔} \end{split}\end{equation}[/latex]
Step 7: Answer the question.
Arianna can make each of the two equal sides 48 inches long.
Exercise 25
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?
Exercise 25 Answer
14 ft
Exercise 26
A boat’s sail is an isosceles triangle with a base of 8 metres. The perimeter is 22 metres. How long is each of the equal sides of the sail?
Exercise 26 Answer
7 m
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 (see Figure 7.1.30).
A trapezoid has a larger base, B, and a smaller base, b. The height h is the distance between the bases.
Formula for the area of a trapezoid:
[latex]{\text{Area}}_{\text{trapezoid}}=\frac{1}{2}h\left(b+B\right)[/latex]
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 7.1.31).
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 7.1.32).
The formula for the area of a trapezoid is:
[latex]Area_{trapezoid} = \dfrac{1}{2}h(\textcolor{teal}{b}+\textcolor{red}{B})[/latex]
If we distribute, we get:
[latex]Area_{trapezoid} = \dfrac{1}{2}\textcolor{teal}{b}h+\dfrac{1}{2}\textcolor{red}{B} \Rightarrow Area_{trapezoid} = A_{\textcolor{teal}{blue\Delta}}+A_{\textcolor{red}{red\Delta}}[/latex]
Properties of trapezoids:
- A trapezoid has four sides.
- Two of its sides are parallel, and two sides are not.
- The area, A, of a trapezoid is [latex]\text{A}=\dfrac{1}{2}h\left(b+B\right)[/latex].
Example N
Find the area of a trapezoid whose height is 6 inches and whose bases are 14 and 11 inches.
Step 1: Read the problem. Draw the figure and label it with the given information (see Figure 7.1.33).
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.
[latex]\begin{equation}\begin{split}A&=\dfrac{1}{2} \cdot h \cdot (b+B) \\ A&=\dfrac{1}{2} \cdot 6 \cdot (11+14) \end{split}\end{equation}[/latex]
Step 5: Solve the equation.
[latex]\begin{equation}\begin{split} A&=\dfrac{1}{2} \cdot 6(25) \\ A&=3(25) \\ A&=25\text{ square inches} \end{split}\end{equation}[/latex]
Step 6: Check. Is this answer reasonable?
If we draw a rectangle around the trapezoid with 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 with the same little base b and a height h, its area should be smaller than that of the trapezoid.
See Figure 7.1.34 for more details.
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.
Exercise 27
The height of a trapezoid is 14 yards, and the bases are 7 and 16 yards. What is the area?
Exercise 27 Answer
161 sq. yd
Exercise 28
The height of a trapezoid is 18 centimetres, and the bases are 17 and 8 centimetres. What is the area?
Exercise 28 Answer
225 sq. cm
Example O
Find the area of a trapezoid whose height is 5 feet and whose bases are 10.3 and 13.7 feet.
Step 1: Read the problem. Draw the figure and label it with the given information (see Figure 7.1.35).
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.
[latex]\begin{equation}\begin{split}A&=\dfrac{1}{2} \cdot h \cdot (b+B) \\ A&=\dfrac{1}{2} \cdot 5 \cdot (10.3+13.7) \end{split}\end{equation}[/latex]
Step 5: Solve the equation.
[latex]\begin{equation}\begin{split} A&=\dfrac{1}{2} \cdot 5(24) \\ A&=12 \cdot 5\\ A&=60\text{ square feet} \end{split}\end{equation}[/latex]
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 (see Figure 7.1.36).
Step 7: Answer the question.
The area of the trapezoid is 60 square feet.
Exercise 29
The height of a trapezoid is 7 centimetres, and the bases are 4.6 and 7.4 centimetres. What is the area?
Exercise 29 Answer
42 sq. cm
Exercise 30
The height of a trapezoid is 9 metres, and the bases are 6.2 and 7.8 metres. What is the area?
Exercise 30 Answer
63 sq. m
Example P
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?
Step 1: Read the problem. Draw the figure and label it with the given information (see Figure 7.1.37).
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.
[latex]\begin{equation}\begin{split}A&=\dfrac{1}{2} \cdot h \cdot (b+B) \\ A&=\dfrac{1}{2} \cdot 3.4 \cdot (5.6+8.2) \end{split}\end{equation}[/latex]
Step 5: Solve the equation.
[latex]\begin{equation}\begin{split} A&=\dfrac{1}{2}(3.4)(13.8) \\ A&=23.46\text{ square yards} \end{split}\end{equation}[/latex]
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 (see Figure 7.1.38).
Step 7: Answer the question.
Vinny has 23.46 square yards in which he can plan.
Exercise 31
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?
Exercise 31 Answer
40.25 sq. yd
Exercise 32
Kira wants to 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?
Exercise 32 Answer
240 sq. ft.
Additional online resources:
- Perimeter of a Rectangle
- Ex: Determine the Perimeter of a Rectangle Involving Whole Numbers [2:41 min] (Mathispower4u, 2011c).
- Area of a Rectangle
- Ex: Determine the Area of a Rectangle Involving Whole Numbers [2:06 min] (Mathispower4u, 2011b).
- Perimeter and Area Formulas
- Perimeter and Area Formulas [9:44 min] (Mathispower4u, 2009).
- Area of a Triangle
- Ex: Area of a Triangle (Whole Numbers) [1:56 min] (Mathispower4u, 2011d).
- Area of a Triangle with Fractions
- Example: Area of a Triangle Involving Fractions [1:34 min] (Mathispower4u, 2011a).
- Area of a Trapezoid
- Ex: Find the Area of a Trapezoid [2:32 min] (Mathispower4u, 2012).
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=L\cdot W[/latex]
Triangle Properties
- For any triangle [latex]\delta ABC[/latex], the sum of the measures of the angles is 180°.
- [latex]m\angle A+m\angle B+m\angle C=180[/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=\dfrac{1}{2}bh[/latex]
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 with four sides and four right angles.
- Trapezoid — A trapezoid is a four-sided figure, a quadrilateral, with two sides that are parallel and two sides that are not.
7.1 Practice Questions
1. Understand Linear, Square, & Cubic Measure
In the following exercises, determine whether you would measure each item using linear, square, or cubic units.
- amount of water in a fish tank
- length of dental floss
- living area of an apartment
- floor space of a bathroom tile
- height of a doorway
- 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.
-
Figure 7.1.39 
2. Use the Properties of Rectangles
In the following exercises, find the a) perimeter and b) area of each rectangle.
- The length of a rectangle is 85 feet, and the width is 45 feet.
- The length of a rectangle is 26 inches, and the width is 58 inches.
- A rectangular room is 15 feet wide by 14 feet long.
- A driveway is in the shape of a rectangle 20 feet wide by 35 feet long.
In the following exercises, solve.
- Find the length of a rectangle with a perimeter of 124 inches and a width of 38 inches.
- Find the length of a rectangle with a perimeter of 124 inches and a width of 38 inches.
- Find the width of a rectangle with a perimeter of 92 metres and a length of 19 metres.
- Find the width of a rectangle with a perimeter of 16.2 metres and a length of 3.2 metres.
- The area of a rectangle is 414 square metres. The length is 18 metres. What is the width?
- The area of a rectangle is 782 square centimetres. The width is 17 centimetres. What is the length?
- The length of a rectangle is 9 inches more than the width. The perimeter is 46 inches. Find the length and the width.
- The width of a rectangle is 8 inches more than the length. The perimeter is 52 inches. Find the length and the width.
- 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.
- The perimeter of a rectangle is 62 feet. The width is 7 feet less than the length. Find the length and the width.
- 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.
- 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.
- The perimeter of a rectangle is 150 feet. The length of the rectangle is twice the width. Find the length and width of the rectangle.
- The length of a rectangle is three times the width. The perimeter is 72 feet. Find the length and width of the rectangle.
- The length of a rectangle is 3 metres less than twice the width. The perimeter is 36 metres. Find the length and width.
- The length of a rectangle is 5 inches more than twice the width. The perimeter is 34 inches. Find the length and width.
- The width of a rectangular window is 24 inches. The area is 624 square inches. What is the length?
- The length of a rectangular poster is 28 inches. The area is 1316 square inches. What is the width?
- The area of a rectangular roof is 2310 square metres. The length is 42 metres. What is the width?
- The area of a rectangular tarp is 132 square feet. The width is 12 feet. What is the length?
- The perimeter of a rectangular courtyard is 160 feet. The length is 10 feet more than the width. Find the length and the width.
- The perimeter of a rectangular painting is 306 centimetres. The length is 17 centimetres more than the width. Find the length and the width.
- 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.
- 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.
3. Use the Properties of Triangles
In the following exercises, solve using the properties of triangles.
- Find the area of a triangle with a base of 12 inches and a height of 5 inches.
- Find the area of a triangle with a base of 45 centimetres and a height of 30 centimetres.
- Find the area of a triangle with a base of 8.3 metres and a height of 6.1 metres.
- Find the area of a triangle with a base of 24.2 feet and a height of 20.5 feet.
- A triangular flag has a base of 1 foot and a height of 1.5 feet. What is its area?
- A triangular window has a base of 8 feet and a height of 6 feet. What is its area?
- If a triangle has sides of 6 feet and 9 feet and the perimeter is 23 feet, how long is the third side?
- If a triangle has sides of 14 centimetres and 18 centimetres and the perimeter is 49 centimetres, how long is the third side?
- What is the base of a triangle with an area of 207 square inches and a height of 18 inches?
- What is the height of a triangle with an area of 893 square inches and a base of 38 inches?
- The perimeter of a triangular reflecting pool is 36 yards. The lengths of the two sides are 10 yards and 15 yards. How long is the third side?
- A triangular courtyard has a perimeter of 120 metres. The lengths of the two sides are 30 metres and 50 metres. How long is the third side?
- An isosceles triangle has a base of 20 centimetres. If the perimeter is 76 centimetres, find the length of each of the other sides.
- An isosceles triangle has a base of 25 inches. If the perimeter is 95 inches, find the length of each of the other sides.
- Find the length of each side of an equilateral triangle with a perimeter of 51 yards.
- Find the length of each side of an equilateral triangle with a perimeter of 54 metres.
- The perimeter of an equilateral triangle is 18 metres. Find the length of each side.
- The perimeter of an equilateral triangle is 42 miles. Find the length of each side.
- 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.
- 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.
- A dish is in the shape of an equilateral triangle. Each side is 8 inches long. Find the perimeter.
- A floor tile is in the shape of an equilateral triangle. Each side is 1.5 feet long. Find the perimeter.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
4. Use the Properties of Trapezoids
In the following exercises, solve using the properties of trapezoids.
- The height of a trapezoid is 12 feet, and the bases are 9 and 15 feet. What is the area?
- The height of a trapezoid is 24 yards, and the bases are 18 and 30 yards. What is the area?
- Find the area of a trapezoid with a height of 51 metres and bases of 43 and 67 metres.
- Find the area of a trapezoid with a height of 62 inches and bases of 58 and 75 inches.
- The height of a trapezoid is 15 centimetres, and the bases are 12.5 and 18.3 centimetres. What is the area?
- The height of a trapezoid is 48 feet, and the bases are 38.6 and 60.2 feet. What is the area?
- Find the area of a trapezoid with a height of 4.2 metres and bases of 8.1 and 5.5 metres.
- Find the area of a trapezoid with a height of 32.5 centimetres and bases of 54.6 and 41.4 centimetres.
- 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?
- Niko wants to tile the floor of his bathroom. The floor is shaped like a trapezoid with a width of 5 feet and lengths of 5 feet and 8 feet. What is the area of the floor?
- Theresa needs a new top for her kitchen counter. The counter is shaped like a trapezoid with a width of 18.5 inches and lengths 62 and 50 inches. What is the area of the counter?
- Elena is knitting a scarf. The scarf will be shaped like a trapezoid with a width of 8 inches and lengths of 48.2 inches and 56.2 inches. What is the area of the scarf?
5. Everyday Math
- Fence: Jose just removed the children’s playset from his backyard 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?
- 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.
- 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?
- Painting: Caleb wants to paint one wall of his attic. The wall is shaped like a trapezoid with a height of 8 feet and bases of 20 feet and 12 feet (like Figure 7.1.39). The cost of painting one square foot of wall is about 0.05. About how much will it cost for Caleb to paint the attic wall?
Figure 7.1.39
6. Writing Exercises
- Look at the two figures in Figure 7.1.40.
Figure 7.1.40 - Which figure looks like it has the larger area? Which looks like it has the larger perimeter?
- Now, calculate the area and perimeter of each figure. Which has the larger area? Which has the larger perimeter?
- The length of a rectangle is 5 feet more than the width. The area is 50 square feet. Find the length and the width.
- Write the equation you would use to solve the problem.
- Why can’t you solve this equation with the methods you learned in the previous chapter?
7.1: Practice Answers
- Understand linear, square, and cubic measure
- cubic
- square
- linear
- 10 cm and 4 sq. cm
- 8 cm, 3 sq. cm
- 10 cm, 5 sq. cm
- Use the properties of rectangles
- 260 ft, 3825 ft
- 58 ft, 210 sq. ft
- 24 inches
- 27 metres
- 23 m
- 7 in., 16 in.
- 17 m, 12 m
- 13.5 m, 12.8 m
- 25 ft, 50 ft
- 7 m, 11 m
- 26 in.
- 55 m
- 35 ft, 45 ft
- 76 in., 36 in.
- Use the properties of triangles
- 30 sq. in.
- 25.315 sq. m
- 0.75 sq. ft
- 8 ft
- 23 in.
- 11 yds.
- 28 cm
- 17 yds.
- 6 m
- 15 ft
- 24 in.
- 27.5 in.
- 12 ft, 13 ft, 14 ft
- 3 ft, 6 ft, 8 ft
- Use the properties of trapezoids
- 144 sq. ft
- 2805 sq. m
- 231 sq. cm
- 28.56 sq. m
- 13.5 sq. ft
- 1036 sq. in.
- Everyday math
- 15 ft
- $240
- Writing exercises
- Answers will vary.
- Answers will vary.
References
Mathispower4u. (2009, December 14). Perimeter and area formulas [Video]. YouTube. https://youtu.be/ZASBmoylCPc?si=qVwtR8CyNr1fWdn4.
Mathispower4u. (2011a, June 11). Example: Area of a triangle involving fractions [Video]. YouTube. https://youtu.be/Xv3tdWcViW0?si=z_lSCL97kDqg7CVi.
Mathispower4u. (2011b, August 5). Ex: Determine the area of a rectangle involving whole numbers [Video]. YouTube. https://youtu.be/TN4tm_rONNc?si=4Q3JhdyBq_g_SzTv.
Mathispower4u. (2011c, August 5). Ex: Determine the perimeter of a rectangle involving whole numbers [Video]. YouTube. https://youtu.be/UuyW51OVPJ0?si=LO7zj0JwoqpJxh6x.
Mathispower4u. (2011d, August 12). Ex: Area of a triangle (whole numbers) [Video]. YouTube. https://youtu.be/yEV8-sc094c?si=mTvISlklKiCxzRNX.
Mathispower4u. (2012, February 24). Ex: Find the area of a trapezoid [Video]. YouTube. https://youtu.be/WNo7s-XoI4w?si=FNeWuG4SAx04MTJM.
Attributions
All figures in this chapter are from 3.2 Use Properties of Rectangles, Triangles, and Trapezoids in Introductory Algebra by Izabela Mazur, via BCcampus.
This chapter has been adapted from 3.2 Use Properties of Rectangles, Triangles, and Trapezoids in Introductory Algebra (BCcampus) by Izabela Mazur (2021), which is under a CC BY 4.0 license.
The original chapter was adapted from 9.4 Use Properties of Rectangles, Triangles, and Trapezoids in Prealgebra 2e (OpenStax) by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis (2020), which is under a CC BY 4.0 license. Adapted by Izabela Mazur.
