GM Lesson 034 Volumes of Pyramids, Cones and Spheres

Learning Intentions

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

• Calculate the volume of a pyramid using base area and perpendicular height.
• Calculate the volume of a cone using radius and perpendicular height.
• Calculate the volume of a sphere using radius.
• Interpret volume answers using appropriate cubic units.

Prerequisites

Students should already be able to:

• Calculate the area of rectangles, triangles and circles.
• Substitute values into a formula.
• Distinguish between radius, diameter, height and slant height.
• Round decimal answers appropriately.
• Use cubic units such as , and .

Key Idea Summary

This lesson extends volume calculations from prisms and cylinders to pyramids, cones and spheres.

A pyramid has volume:

where is the area of the base and is the perpendicular height.

A cone has volume:

where is the radius and is the perpendicular height.

A sphere has volume:

where is the radius.

The key comparison is:

• A pyramid has one-third the volume of a prism with the same base area and perpendicular height.
• A cone has one-third the volume of a cylinder with the same radius and perpendicular height.
• A sphere volume depends only on the radius.

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

Teacher Explanation

For pyramids and cones, the height must be the perpendicular height. This is the vertical distance from the apex to the base.

The slant height is not used for volume. Slant height is used for surface area.

For a pyramid:

For a cone:

For a sphere:

When calculating volume:

1. Identify the correct solid.
2. Identify the required measurements.
3. Substitute values into the formula.
4. Calculate carefully.
5. Write the answer using cubic units.

Worked Example 1: Volume of a Rectangular-Based Pyramid

A pyramid has a rectangular base measuring by . Its perpendicular height is . Find its volume.

The formula is:

First calculate the base area:

Substitute into the volume formula:

Therefore, the volume is:

Worked Example 2: Volume of a Triangular-Based Pyramid

A pyramid has a triangular base with base length and perpendicular height . The perpendicular height of the pyramid is . Find the volume.

The formula is:

First calculate the area of the triangular base:

Now use the pyramid volume formula:

Therefore, the volume is:

Worked Example 3: Volume of a Cone

A cone has radius and perpendicular height . Find its volume correct to one decimal place.

The formula is:

Substitute and :

Therefore, the volume is approximately:

Worked Example 4: Volume of a Cone Given Diameter

A traffic cone is modelled as a cone with diameter and perpendicular height . Find its volume to the nearest cubic centimetre.

The radius is half the diameter:

Use the cone volume formula:

Substitute and :

Therefore, the volume is approximately:

Worked Example 5: Volume of a Sphere

A sphere has radius . Find its volume correct to one decimal place.

The formula is:

Substitute :

Therefore, the volume is approximately:

Worked Example 6: Comparing a Cone and a Cylinder

A cone and a cylinder have the same radius and the same perpendicular height .

Find the volume of each solid and compare them.

Cylinder:

Cone:

The cone has one-third the volume of the cylinder because it has the same base and perpendicular height.

Understanding Checks

Check 1

A pyramid has base area and perpendicular height .

Which formula should be used?

Check 2

A cone has diameter .

What radius should be used in the formula?

Check 3

A cone has slant height and perpendicular height .

Which height is used for volume?

The perpendicular height, , is used.

Check 4

A sphere has radius .

Which expression correctly represents its volume?

Check 5

A pyramid and a prism have the same base area and perpendicular height.

How are their volumes related?

The pyramid has one-third the volume of the prism.

Exercises

Simple Familiar Exercises

Exercise 1

Find the volume of a pyramid with base area and perpendicular height .

Exercise 2

Find the volume of a pyramid with rectangular base by and perpendicular height .

Exercise 3

Find the volume of a cone with radius and perpendicular height . Give your answer correct to one decimal place.

Exercise 4

Find the volume of a cone with radius and perpendicular height . Give your answer correct to one decimal place.

Exercise 5

Find the volume of a sphere with radius . Give your answer correct to one decimal place.

Exercise 6

Find the volume of a sphere with radius . Give your answer correct to one decimal place.

Complex Familiar Exercises

Exercise 7

A pyramid has a triangular base with base length and perpendicular height . The perpendicular height of the pyramid is . Find the volume.

Exercise 8

A cone has diameter and perpendicular height . Find the volume correct to the nearest cubic centimetre.

Exercise 9

A sphere has diameter . Find its volume correct to the nearest cubic centimetre.

Exercise 10

A square-based pyramid has base side length and perpendicular height . Find its volume.

Exercise 11

A cone and a cylinder have the same radius and perpendicular height .

Find the volume of the cylinder and the volume of the cone. Explain the relationship between the two answers.

Exercise 12

A spherical ball has radius . A smaller spherical ball has radius .

Find the volume of each ball correct to one decimal place. How many times larger is the volume of the larger ball?

Homework Problems

Homework 1

Find the volume of a pyramid with base area and perpendicular height .

Homework 2

Find the volume of a rectangular-based pyramid with base dimensions by and perpendicular height .

Homework 3

Find the volume of a cone with radius and perpendicular height . Give your answer correct to one decimal place.

Homework 4

Find the volume of a cone with diameter and perpendicular height . Give your answer correct to one decimal place.

Homework 5

Find the volume of a sphere with radius . Give your answer correct to one decimal place.

Homework 6

Find the volume of a sphere with diameter . Give your answer correct to one decimal place.

Homework 7

A cone-shaped party hat has diameter and perpendicular height . Find its volume correct to the nearest cubic centimetre.

Homework 8

A square-based pyramid has base side length and perpendicular height . Find its volume.

Homework 9

A composite solid is made from a cone joined to a hemisphere. Both have radius . The cone has perpendicular height .

Find the total volume correct to one decimal place.

Homework 10

A sphere has volume approximately .

Use

to determine the radius of the sphere.

Next: GM Lesson 035 Capacity and Mixed Measurement Problems