GM Lesson 033 Volumes of Prisms and Cylinders

Learning Intentions

By the end of this lesson, students will be able to:

• Calculate the volume of a prism using .
• Calculate the volume of a cylinder using .
• Interpret volume calculations using cubic units.

Prerequisites

Students should already be able to:

• Calculate the area of rectangles, triangles and circles.
• Identify the base and perpendicular height of a three-dimensional object.
• Substitute values into formulae.
• Round answers appropriately.
• Use square units for area and cubic units for volume.

Key Idea Summary

Volume measures the amount of space inside a three-dimensional object.

For any prism:

where:

• is volume
• is the area of the base
• is the perpendicular height or length of the prism

For a cylinder:

where:

• is volume
• is the radius of the circular base
• is the perpendicular height of the cylinder

Volume is measured in cubic units, such as , or .

A cylinder is a type of prism-like solid because its cross-section remains the same all the way through, but its base is circular.

Direct Instruction and Worked Examples

Time Allocation

Time Allocation

• Introduction, warmup and vocabulary: 5 minutes
• Direct instruction: 15 minutes
• Understanding checks: 5 minutes
• Exercises: 20 minutes
• Homework: 20 to 30 minutes outside the lesson it was taught in.

Link to original

Direct Instruction

To calculate the volume of a prism, first identify the base shape. Then calculate the area of the base and multiply by the perpendicular height.

The logic is:

So, if the base area is and the prism extends for height , then:

For a cylinder, the base is a circle. Since the area of a circle is:

the volume formula becomes:

Worked Example 1: Volume of a Rectangular Prism

A storage box is long, wide and high. Find its volume.

The base is a rectangle.

Now use:

Therefore, the volume of the storage box is:

Worked Example 2: Volume of a Triangular Prism

A triangular prism has a triangular base with base length and perpendicular height . The prism is long. Find its volume.

First calculate the area of the triangular base.

Now use:

The prism length is the perpendicular height of the solid.

Therefore, the volume of the triangular prism is:

Worked Example 3: Volume of a Cylinder

A cylindrical water tank has radius and height . Find its volume correct to decimal places.

Use:

Substitute and .

Therefore, the volume of the water tank is approximately:

Worked Example 4: Choosing the Correct Formula

A solid has the same trapezium-shaped cross-section all the way through. The trapezium has parallel sides of and , with perpendicular height . The solid is long. Find the volume.

Since the cross-section is the same all the way through, the solid is a prism.

Use:

First find the area of the trapezium base.

Now multiply by the length of the prism.

Therefore, the volume is:

Understanding Checks

Understanding Check 1

A rectangular prism has dimensions , and .

1. What is the area of the rectangular base?
2. What formula should be used for the volume?
3. What is the volume?

Expected answer:

Understanding Check 2

A cylinder has radius and height .

1. What is the value of ?
2. What is the value of ?
3. What expression gives the volume?

Expected answer:

Understanding Check 3

A prism has base area and height .

Find its volume.

Expected answer:

Understanding Check 4

Explain why the answer to a volume problem should use cubic units instead of square units.

Expected response:

Volume measures three-dimensional space, so the units must describe length, width and height. Therefore, volume uses cubic units such as or .

Exercises

Simple Familiar Exercises

Exercise 1

Find the volume of a rectangular prism with length , width and height .

Exercise 2

Find the volume of a rectangular prism with base area and height .

Exercise 3

Find the volume of a triangular prism with triangular base length , triangular height and prism length .

Exercise 4

Find the volume of a cylinder with radius and height . Give your answer in terms of .

Exercise 5

Find the volume of a cylinder with radius and height . Give your answer correct to decimal places.

Complex Familiar Exercises

Exercise 6

A concrete slab is shaped like a rectangular prism. It is long, wide and thick.

Find the volume of concrete required.

Exercise 7

A triangular prism has a right-angled triangular base with perpendicular side lengths and . The prism is long.

Find the volume of the prism.

Exercise 8

A cylindrical pipe has internal radius and length .

Find the internal volume of the pipe correct to the nearest cubic centimetre.

Exercise 9

A prism has a parallelogram-shaped base. The base of the parallelogram is and its perpendicular height is . The prism has length .

Find the volume of the prism.

Exercise 10

A cylindrical rainwater tank has diameter and height .

Find the volume of the tank correct to decimal places.

Homework Problems

Homework 1

Find the volume of a rectangular prism with dimensions , and .

Homework 2

A prism has base area and perpendicular height . Find its volume.

Homework 3

Find the volume of a triangular prism with triangular base length , triangular height and prism length .

Homework 4

Find the volume of a cylinder with radius and height . Give your answer correct to the nearest cubic centimetre.

Homework 5

A swimming pool is shaped like a rectangular prism. It is long, wide and deep.

Find the volume of water needed to fill the pool.

Homework 6

A cylindrical grain silo has diameter and height .

Find its volume correct to decimal place.

Homework 7

A prism has a trapezium-shaped base with parallel sides and and perpendicular height . The prism is long.

Find the volume of the prism.

Homework 8

A cylindrical container has volume and radius . Use to find its height.

Next: GM Lesson 034 Volumes of Pyramids, Cones and Spheres