GM Lesson 031 Surface Area of Pyramids and Cones

Learning Intentions

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

• Identify the faces required for the surface area of a pyramid.
• Calculate the surface area of a cone using radius and slant height.
• Solve practical surface area problems involving pyramids and cones.

Prerequisites

Students should already be able to:

• Calculate the area of rectangles, squares, triangles and circles.
• Use the circle area formula:

• Use the triangle area formula:

• Distinguish between perpendicular height and slant height.
• Round answers appropriately and use square units for surface area.

Key Idea Summary

Surface area is the total area of the outside surfaces of a three-dimensional object.

For a pyramid, the surface area is found by adding:

For a square pyramid with base side length and slant height :

or equivalently:

For a cone with radius and slant height :

where:

• is the area of the circular base.
• is the curved surface area.
• is the radius of the circular base
• is the slant height (and radius of the sector component)

The perpendicular height of a cone or pyramid is not the same as the slant height. If the perpendicular height is given, then the slant length should be found first using and .

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

A pyramid has one base and several triangular faces. To calculate its surface area, identify the shape of the base, calculate the base area, then calculate the area of each triangular face.

A cone has one circular base and one curved surface. Its total surface area is calculated using:

The value is the slant height, measured along the outside of the cone from the rim of the base to the vertex.

Worked Example 1: Surface Area of a Square Pyramid

A square pyramid has base side length and slant height . Calculate its surface area.

The base is a square:

Each triangular face has base and height :

There are triangular faces:

Therefore:

Worked Example 2: Surface Area of a Rectangular Pyramid

A rectangular pyramid has a base measuring by . The two triangular faces with base have slant height . The two triangular faces with base have slant height . Calculate the surface area.

Base area:

Two larger triangular faces:

Two smaller triangular faces:

Total surface area:

Worked Example 3: Surface Area of a Cone

A cone has radius and slant height . Calculate its surface area, correct to decimal place.

Use:

Substitute and :

Worked Example 4: Practical Cone Problem

A party hat is shaped like a cone with radius and slant height . The circular base is open, so only the curved surface is covered with cardboard. Calculate the area of cardboard needed.

Since the base is open, use only the curved surface area:

Substitute and :

The area of cardboard needed is approximately:

Understanding Checks

Check 1

A pyramid has a square base and triangular faces. What areas must be added to find the total surface area?

Expected response:

Check 2

A cone has radius and slant height . Which expression gives the total surface area?

A. B. C. D.

Correct response:

Check 3

Explain why the perpendicular height of a cone cannot be used directly in the formula:

Expected response:

The formula uses , the slant height. The slant height is measured along the outside curved surface of the cone, while the perpendicular height is measured vertically through the centre.

Exercises

Simple Familiar Exercises

Exercise 1

A square pyramid has base side length and slant height . Calculate its total surface area.

Exercise 2

A square pyramid has base side length and slant height . Calculate its total surface area.

Exercise 3

A cone has radius and slant height . Calculate its total surface area, correct to decimal place.

Exercise 4

A cone has radius and slant height . Calculate its total surface area, correct to decimal place.

Complex Familiar Exercises

Exercise 5

A rectangular pyramid has a base measuring by . The two triangular faces with base have slant height . The two triangular faces with base have slant height . Calculate the total surface area.

Exercise 6

A cone has radius and slant height . The circular base is not included because the cone is open at the bottom. Calculate the curved surface area, correct to decimal place.

Exercise 7

A solid cone has radius and slant height . Calculate its total surface area, correct to decimal place.

Exercise 8

A square pyramid has total surface area . Its base side length is and its slant height is unknown. Find the slant height.

Homework Problems

Homework 1

A square pyramid has base side length and slant height . Calculate its total surface area.

Homework 2

A square pyramid has base side length and slant height . Calculate its total surface area.

Homework 3

A cone has radius and slant height . Calculate its total surface area, correct to decimal place.

Homework 4

A cone has radius and slant height . Calculate its curved surface area only, correct to decimal place.

Homework 5

A rectangular pyramid has a base measuring by . The triangular faces with base have slant height . The triangular faces with base have slant height . Calculate the total surface area.

Homework 6

A cone-shaped cover has radius and slant height . The circular base is open. Calculate the area of material needed, correct to the nearest square centimetre.

Next: GM Lesson 032 Surface Area of Spheres and Composite Solids