GM Lesson 029 Surface Area of Rectangular Prisms

Learning Intentions

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

• Identify all faces of a rectangular prism.
• Calculate the surface area of a rectangular prism.
• Interpret surface area in practical contexts using square units.

Prerequisites

Students should already be able to:

• Calculate the area of a rectangle using .
• Identify length, width and height in three-dimensional objects.
• Use correct square units such as , and .
• Add several area values to find a total area.

Key Idea Summary

A rectangular prism has rectangular faces.

Opposite faces are equal in area.

For a rectangular prism with length , width and height :

Equivalently:

Surface area measures the total area covering the outside of a three-dimensional object, so the answer is written in square units.

Examples of surface area contexts include:

• wrapping a box
• painting a rectangular tank
• covering a package with cardboard
• tiling or laminating the outside of a rectangular object

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 rectangular prism has three different pairs of opposite faces:

• top and bottom: each has area
• front and back: each has area
• left and right sides: each has area

Therefore:

So:

Students should follow this process:

1. Identify , and .
2. Find the area of each pair of opposite faces.
3. Add the areas of all faces.
4. State the answer using square units.

Worked Example 1: Surface Area from Given Dimensions

A rectangular prism has length , width and height . Find its surface area.

Identify:

Use:

Substitute:

Calculate:

Therefore, the surface area is:

Worked Example 2: Surface Area Using a Net

A rectangular box has dimensions by by .

Find the area of each pair of opposite faces.

Top and bottom:

Front and back:

Left and right sides:

Add:

Therefore, the surface area is:

Worked Example 3: Practical Context

A rectangular garden storage box is long, wide and high. It needs to be painted on all outside faces. Find the total area to be painted.

Identify:

Use:

Substitute:

Calculate:

Therefore, the area to be painted is:

Worked Example 4: Missing Surface in Context

A rectangular fish tank is open at the top. Its length is , width is and height is . Find the outside glass area, excluding the open top.

The full surface area would include faces, but the top is missing.

Bottom:

Front and back:

Left and right sides:

Add only the exposed glass faces:

Therefore, the outside glass area is:

Understanding Checks

Check 1

A rectangular prism has dimensions , and .

Which expression correctly finds its surface area?

A.

B.

C.

D.

Check 2

A box has dimensions by by .

Find the area of the top and bottom faces combined.

Check 3

A rectangular prism has dimensions , and .

Complete:

Check 4

Explain why surface area is measured in square units rather than cubic units.

Check 5

A rectangular box has no lid. Which face should be excluded when calculating the material needed to make the box?

Exercises

Simple Familiar Exercises

Exercise 1

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

Exercise 2

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

Exercise 3

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

Exercise 4

A rectangular prism has dimensions , and . Calculate its surface area.

Exercise 5

A rectangular prism has length , width and height . Calculate its surface area.

Complex Familiar Exercises

Exercise 6

A cardboard box has length , width and height . Find the total area of cardboard needed to make the outside of the box.

Exercise 7

A rectangular storage crate is long, wide and high. Calculate its surface area.

Exercise 8

A rectangular prism has dimensions by by . Calculate:

a. the area of the top and bottom faces combined

b. the area of the front and back faces combined

c. the area of the two side faces combined

d. the total surface area

Exercise 9

A rectangular fish tank is open at the top. It has length , width and height . Calculate the area of glass required, excluding the top.

Exercise 10

A rectangular metal container has dimensions , and . It is painted on all outside faces. Paint covers per litre. Determine whether litres of paint is enough.

Homework Problems

Problem 1

Find the surface area of a rectangular prism with dimensions , and .

Problem 2

Find the surface area of a rectangular prism with dimensions , and .

Problem 3

A cereal box has length , width and height . Calculate the outside surface area of the box.

Problem 4

A lidless rectangular container has length , width and height . Calculate the area of material needed to make the container.

Problem 5

A rectangular room is long, wide and high. The walls and ceiling are to be painted, but not the floor.

Calculate the total area to be painted.

Problem 6

A rectangular prism has length , width and height .

Write an expression for its surface area in terms of .

Problem 7

A rectangular prism has dimensions , and .

Another rectangular prism has dimensions , and .

Which prism has the larger surface area, and by how much?

Problem 8

A rectangular box has surface area , length and width . Find the height of the box.

GM Lesson 030 Surface Area of Cylinders