GM Lesson 023 Simple Applications in Three Dimensions

Learning Intentions

By the end of this 45 minute lesson, students will be able to:

• Identify right-angled triangles within simple three-dimensional objects.
• Use Pythagoras’ theorem to calculate diagonal lengths in three-dimensional contexts.
• Solve practical problems involving two connected applications of Pythagoras’ theorem.

Prerequisites

Students should already be able to:

• Identify the hypotenuse in a right-angled triangle.
• Use Pythagoras’ theorem in the form:

where is the hypotenuse and and are the two perpendicular side lengths.

• Rearrange Pythagoras’ theorem to find an unknown shorter side.
• Round decimal answers appropriately.
• Interpret lengths using correct units.

Key Idea Summary

Pythagoras’ theorem can be used in three-dimensional contexts when a right-angled triangle can be identified.

In a rectangular prism, there are two important types of diagonals:

• A face diagonal, which lies across one rectangular face.
• A space diagonal, which passes through the inside of the prism from one corner to the opposite corner.

To find a space diagonal of a rectangular prism, we usually use Pythagoras’ theorem twice.

First, find a face diagonal:

Then, use that diagonal with the height to find the space diagonal:

Equivalently, for a rectangular prism:

However, students should understand this as two connected right-angled triangles rather than simply memorising a new formula.

Direct Instruction and Worked Examples

Introduction: Finding Right Triangles in Three Dimensions

Time: 5 minutes

Explain that Pythagoras’ theorem still only works on right-angled triangles. In three-dimensional problems, the main skill is identifying where the right-angled triangle is hidden.

Common three-dimensional objects include:

• Rectangular prisms
• Boxes
• Rooms
• Cubes
• Ladders or rods placed inside rectangular spaces
• Diagonal braces across rectangular frames

For a rectangular prism with length , width and height , a rod from one bottom corner to the opposite top corner forms a space diagonal. It is not found directly from a single visible rectangle unless the correct right triangle is identified.

Worked Example 1: Face Diagonal of a Rectangular Prism

Time: 7 minutes

A rectangular box has length and width . Find the diagonal across the base of the box.

The base forms a right-angled triangle with perpendicular sides and .

Use:

Substitute:

The diagonal across the base is:

Worked Example 2: Space Diagonal Using Two Applications of Pythagoras’ Theorem

Time: 10 minutes

A rectangular box has length , width and height . Find the length of a rod that fits from one bottom corner to the opposite top corner.

First find the diagonal across the base.

From Worked Example 1:

Now use the base diagonal and the height to form a second right-angled triangle.

The perpendicular sides are and .

Use:

Substitute:

The rod must be approximately:

Worked Example 3: Practical Three-Dimensional Context

Time: 8 minutes

A rectangular storage room is long, wide and high. A cable is to be stretched in a straight line from one bottom corner of the room to the opposite top corner. Find the length of cable required, correct to one decimal place.

First find the diagonal across the floor:

Now use this floor diagonal with the height:

The cable must be approximately:

Worked Example 4: Cube Diagonal

Time: 5 minutes

A cube has side length . Find the space diagonal from one corner to the opposite corner.

First find the diagonal across one square face:

Now use this face diagonal and the third edge of the cube:

The space diagonal is approximately:

Understanding Checks

Time: 5 minutes

Check 1

A rectangular prism has length , width and height .

Which two lengths should be used first to find the diagonal across the base?

Check 2

A student calculates the base diagonal of a rectangular prism as . The prism has height .

Which equation should be used to find the space diagonal?

Check 3

A box is long, wide and high.

Explain why the answer for the space diagonal must be longer than .

Check 4

A cube has side length .

Why are two applications of Pythagoras’ theorem needed to find the space diagonal?

Exercises

Time: 10 minutes

Simple Familiar Exercises

Exercise 1

A rectangular prism has length and width .

Find the diagonal across the rectangular base.

Exercise 2

A rectangular prism has length and width .

Find the diagonal across the floor.

Exercise 3

A cube has side length .

Find the diagonal across one square face.

Exercise 4

A rectangular box has base diagonal and height .

Find the space diagonal.

Complex Familiar Exercises

Exercise 5

A rectangular box has length , width and height .

Find the space diagonal, correct to one decimal place.

Exercise 6

A room is long, wide and high.

Find the length of a straight cable from one bottom corner to the opposite top corner, correct to one decimal place.

Exercise 7

A cube has side length .

Find the space diagonal, correct to one decimal place.

Exercise 8

A rectangular prism has length , width and height .

Find the diagonal from one bottom corner to the opposite top corner.

Homework Problems

Homework should take no more than 30 minutes.

Problem 1

A rectangular box has length , width and height .

Find the space diagonal.

Problem 2

A classroom is long, wide and high.

Find the length of a straight line from one floor corner to the opposite ceiling corner, correct to one decimal place.

Problem 3

A cube has side length .

Find:

a. the diagonal across one face

b. the space diagonal of the cube

Give answers correct to one decimal place where necessary.

Problem 4

A rectangular prism has length , width and height .

Find the space diagonal, correct to one decimal place.

Problem 5

A tent has a rectangular floor that is long and wide. The highest point of the tent is directly above one corner and is above the ground.

Find the straight-line distance from the highest point to the opposite floor corner, correct to one decimal place.

Problem 6

A rectangular prism has length and width . Its space diagonal is .

Find the height of the prism.

Next: GM Lesson 024 Perimeters of Standard Two-Dimensional Objects