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4.8 Volume of a Right Cylinder 215
Develop and use a formula to find the volume of a right cylinder.
Focus
Volume of a Right Cylinder
Work with a partner. You will need 2 identical rectangular sheets of construction paper, rice, and tape.
Roll one sheet of paper lengthwise to create a tube. Tape the edges together. Repeat with the second sheet of paper. This time roll the paper widthwise. Predict which tube has the greater volume. Use rice to check your prediction. Do the results match your prediction? Explain. Calculate the volume of the taller tube. How did you use the diameter and radius in your calculations? How did you use ?
Here is a way to visualize a right cylinder. A circle is translated through the air so that the circle is always parallel to its original position.
How does this relate to the triangular prism in Lesson 4.6, page 202?
Share your strategy for calculating the volume with another pair of classmates. Work together to write a formula for the volume of a right cylinder. Use any of diameter, radius, height, and in your formula. Use your formula to find the volume of the shorter tube.
Investigate
Reflect
Share &
216 UNIT 4: Measuring Prisms and Cylinders
The volume of a right prism is: base area height We can use this formula to find the volume of a right cylinder.
Example 1
The area of the base of a cylinder is about 154 cm^2. The height of the cylinder is 24 cm. Find the volume of the cylinder.
A Solution Volume of a cylinder base area height 154 24 3696
The volume of the cylinder is about 3696 cm^3.
We can write an algebraic formula for the volume. The base of a cylinder is a circle with radius r. The area of a circle is: A r^2 Let the height of the cylinder be h.
So, the volume of a cylinder is: V base area height area of circle height r^2 h r^2 h
So, a formula for the volume of a cylinder is V r^2 h , where r is the radius of its base, and h its height.
Connect
h r A = r 2
24 cm
A =⋅ 154 cm^2
218 UNIT 4: Measuring Prisms and Cylinders
Check
Give each volume to the nearest cubic unit.
- The base area and height of each cylinder are given to one decimal place. Calculate the volume of each cylinder. a) b)
c)
- Calculate the volume of each cylinder. a) b) c)
- A candle mould is cylindrical. Its radius is 5 cm and its height is 20 cm. What volume of wax will fit in the mould?
Apply
- Find a right cylinder in the classroom. a) Measure its height and diameter. b) Calculate its base area. c) Calculate its volume. 8. A hockey puck is a solid piece of rubber with the dimensions shown. How much rubber is used to make a hockey puck? 9. How do the volumes of these cylinders compare? How can you tell without calculating each volume?
- Kari has 125 mL of water. She wants to pour it into one of these cylindrical bottles. Which bottle will hold all the water? How do you know? Bottle A: d = 7 cm, h = 3 cm Bottle B: r = 2 cm, h = 6 cm Bottle C: r = 3.5 cm, h = 7 cm Bottle D: d = 3 cm, h = 4 cm
- Assessment Focus Frozen apple juice comes in cylindrical cans. A can is 12 cm high with radius 3.5 cm. a) What is the capacity of the can? b) What happens to the capacity of the can if the dimensions of the radius and height are switched? Why does this happen?
Practice
10 cm
4 cm
50 mm
15 mm
12.4 m
2.9 m
10 cm
2.5 cm
10.0 cm
A = 78.5 cm^2
5.0 cm
A = 12.6 cm^2
8 cm
A = 201.1 cm^2
6.8 cm
2.4 cm
6.8 cm
2.4 cm
- A core sample of soil is cylindrical. The length of the core is 300 mm. Its diameter is 15 cm. Calculate the volume of soil.
- Carol and Tom are drilling a well for water at their cottage in Lac La Hache, B.C. The drill is about 15 cm wide. Carol and Tom found water at a depth of 25 m. About how much soil did they remove before they found water?
- A farmer has 3 cylindrical containers to hold feed. Each container has radius 91 cm and height 122 cm. What is the total volume of the three containers? How did you find out?
- Orange juice concentrate is poured into cylindrical cans with diameter 7 cm and height 12 cm. A space of 1.5 cm is left at the top of the can to allow for expansion when the concentrate freezes. What volume of concentrate is poured into each can?
- Take It Further Which right cylinder do you think has the greater volume?
- a cylinder with radius 1 m and height 2 m, or - a cylinder with radius 2 m and height 1 m How can you find out without using a calculator? Explain.
- Take It Further A concrete column in a parkade is cylindrical. The column is 10 m high with diameter 3.5 m. a) What is the volume of concrete in one column? b) There are 127 columns in the parkade. What is the total volume of concrete? c) Suppose the concrete in part a is made into a cube. What would the dimensions of the cube be?
- Take It Further A study shows that consumers think the diameter of a large can of coffee is too wide. The study suggests that a narrower can would increase sales. The original can has diameter 20 cm and height 18 cm. Suppose the diameter of the can is decreased by 20% without changing the volume. What is the height of the new can?
4.8 Volume of a Right Cylinder 219
Reflect How did your knowledge of circles help you in this lesson?
300 mm
15 cm
