222. Total Surface Area of Right Prisms
Learning Intentions
- Identify all faces required for total surface area.
- Calculate surface area by adding face areas.
- Interpret surface area Use square units.
Pre-requisite Summary
- A right prism has two congruent, parallel bases.
- The side faces of a right prism are rectangles.
- Total surface area is the sum of the areas of all outside faces.
- Opposite faces of a rectangular prism are congruent.
- The area of a rectangle is
. - The area of a triangle is
. - Surface area is measured in square units such as
, and . - A net can help show every face that must be included.
Worked Examples
Worked Example 1
A rectangular prism has dimensions:
a) Identify all six faces.
b) Pair the congruent faces.
c) State which face areas are needed for total surface area.
Worked Example 2
A triangular prism has triangular bases and three rectangular side faces.
The triangular base has base length
a) Identify the two triangular faces.
b) Identify the three rectangular faces.
c) State which measurements are needed for total surface area.
Worked Example 3
Find the total surface area of a rectangular prism with:
Worked Example 4
Find the total surface area of a cube with side length
Worked Example 5
A triangular prism has a triangular base with base length
Find the total surface area.
Worked Example 6
A cardboard box is shaped like a rectangular prism with length
a) Find the total surface area.
b) Interpret what the answer means in context.
c) State the correct square unit.
Problems
Problem 1
A rectangular prism has dimensions:
a) Identify all six faces.
b) Pair the congruent faces.
c) State which face areas are needed for total surface area.
Problem 2
A triangular prism has triangular bases and three rectangular side faces.
The triangular base has base length
a) Identify the two triangular faces.
b) Identify the three rectangular faces.
c) State which measurements are needed for total surface area.
Problem 3
Find the total surface area of a rectangular prism with:
Problem 4
Find the total surface area of a cube with side length
Problem 5
A triangular prism has a triangular base with base length
Find the total surface area.
Problem 6
A storage box is shaped like a rectangular prism with length
a) Find the total surface area.
b) Interpret what the answer means in context.
c) State the correct square unit.
Exercises
Understanding and Fluency
Exercise 1
A rectangular prism has dimensions:
a) Identify all six faces.
b) Pair the congruent faces.
c) State the three different face areas needed.
Exercise 2
A rectangular prism has length
a) Find the area of the front face.
b) Find the area of the top face.
c) Find the area of the side face.
Exercise 3
Find the total surface area of a rectangular prism with:
Exercise 4
Find the total surface area of a rectangular prism with:
Exercise 5
Find the total surface area of a cube with side length
Exercise 6
Find the total surface area of a cube with side length
Exercise 7
A triangular prism has a triangular base with base length
a) Find the area of one triangular base.
b) Find the combined area of the two triangular bases.
c) Find the total area of the three rectangular faces.
Exercise 8
A triangular prism has a triangular base with side lengths
a) Find the area of one triangular base.
b) Find the combined area of the two triangular bases.
c) Find the total surface area.
Exercise 9
Choose the correct unit for the surface area of each object.
a) A box measured in centimetres
b) A room measured in metres
c) A small plastic piece measured in millimetres
Exercise 10
Complete each statement.
a) If lengths are measured in
b) If lengths are measured in
c) If lengths are measured in
Reasoning
Exercise 11
Explain why total surface area requires all outside faces to be included.
Exercise 12
A student finds only the area of the top face of a rectangular prism and says this is the total surface area.
Explain the error.
Exercise 13
A student gives the surface area of a prism as
Explain why the unit is incorrect.
Exercise 14
Explain why a net can help prevent missing a face when calculating total surface area.
Exercise 15
A triangular prism has two triangular faces and three rectangular faces.
Explain why calculating only the rectangular faces does not give the total surface area.
Problem-solving
Exercise 16
A gift box is shaped like a rectangular prism with length
a) Find the area of each pair of congruent faces.
b) Find the total surface area.
c) Explain what this value represents if the box is wrapped in paper.
Exercise 17
A tent is shaped like a triangular prism. The triangular ends have base length
a) Find the combined area of the two triangular ends.
b) Find the total area of the three rectangular faces.
c) Find the total surface area of the tent.
Exercise 18
A wooden block is a cube with side length
a) Find the area of one face.
b) Find the total surface area.
c) Explain why the answer uses
Exercise 19
A rectangular prism has total surface area
a) Write an expression for the total surface area.
b) Substitute the known values.
c) Find the height
Potential Misunderstandings
- Counting only visible faces instead of all outside faces.
- Forgetting that opposite faces of a rectangular prism are congruent.
- Confusing surface area with volume.
- Using cubic units instead of square units.
- Omitting the two bases of a prism.
- Counting a face twice or missing a face when using a diagram.
- Using the slanted side of a triangle as the perpendicular height.
- Forgetting to multiply by
for pairs of congruent faces. - Assuming all prisms have rectangular bases.
- Not recognising that a cube has six congruent square faces.