125e. Nets and Surface Area of Prisms

Learning Intentions

• To know the meaning of the terms prism, net and total surface area (TSA)
• Solve the total surface area of a prism

Pre-requisite Summary

• Know that a three-dimensional solid has faces, edges and vertices
• Understand that a prism has two matching, parallel bases
• Be able to Identify common 2D shapes such as rectangles, triangles and other polygons
• Know that area measures the amount of surface inside a 2D shape
• Be able to find the area of rectangles and triangles
• Understand that square units such as and are used for surface area
• Be able to add several areas together accurately

Worked Examples

Worked Example 1

State the meaning of each term:

a) prism

b) net

c) total surface area

Worked Example 2

Identify the faces in each prism and Describe its net:

a) a rectangular prism

b) a triangular prism

Worked Example 3

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

Worked Example 4

Find the total surface area of a cube with side length .

Worked Example 5

Find the total surface area of a triangular prism with:

a) triangular ends of base and height

b) prism length

c) the other two sides of the triangular base both

Worked Example 6

A prism has total surface area made from:

a) two congruent end faces of area each

b) three rectangular side faces of areas , and

Find the TSA.

Problems

Problem 1

State the meaning of each term:

a) prism

b) net

c) total surface area

Problem 2

Identify the faces in each prism and describe its net:

a) a cube

b) a pentagonal prism

Problem 3

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

Problem 4

Find the total surface area of a cube with side length .

Problem 5

Find the total surface area of a triangular prism with:

a) triangular ends of base and height

b) prism length

c) the other two sides of the triangular base both

Problem 6

A prism has total surface area made from:

a) two congruent end faces of area each

b) four rectangular side faces of areas , , and

Find the TSA.

Exercises

Understanding and Fluency

Exercise 1.

Complete each statement:

a) A prism has two ______ bases

b) A net is a ______ pattern that can fold to make a solid

c) Total surface area is the ______ area of all faces

Exercise 2.

State whether each statement is true or false:

a) A prism has curved surfaces

b) A net shows all the faces of a solid laid flat

c) TSA is measured in square units

d) A cube is a type of prism

Exercise 3.

Find the total surface area of each rectangular prism:

a)

b)

c)

Exercise 4.

Find the total surface area of each cube:

a) side length

b) side length

c) side length

Exercise 5.

Find the area of each net, then state the TSA of the prism:

a) two ends of area each, and side faces of area , ,

b) two ends of area each, and side faces of area , ,

Exercise 6.

Find the total surface area of each triangular prism:

a) triangular end area , and side rectangles of areas , ,

b) triangular end area , and side rectangles of areas , ,

Exercise 7.

A prism has the following face areas. Find the TSA:

a)

b)

Exercise 8.

Solve each:

a) A rectangular prism has length , width and height . Find its TSA

b) A cube has TSA . Find the area of one face

c) A prism has two end faces of area and lateral faces totalling . Find the TSA

Reasoning

Exercise 9.

Explain why a net is useful for finding the total surface area of a prism.

Exercise 10.

A student says the TSA of a prism is found by adding only the areas of the two end faces. Explain the mistake.

Exercise 11.

Noah says that total surface area should be measured in cubic units because a prism is three-dimensional. Is he correct? Explain.

Exercise 12.

Explain why a cube can be treated as a special rectangular prism when finding TSA.

Exercise 13.

A student adds the edge lengths of a prism instead of the face areas. Describe the error.

Problem-solving

Exercise 14.

A shoebox is a rectangular prism with dimensions , and . Find its total surface area.

Exercise 15.

A cube-shaped gift box has side length . Find its total surface area.

Exercise 16.

A triangular prism has two triangular ends of area each, and three rectangular side faces of areas , and . Find the TSA.

Exercise 17.

A prism net has face areas . Find the total surface area.

Exercise 18.

A rectangular water tank has length , width and height . Find its TSA.

Exercise 19.

A triangular prism has triangular ends each with area , and the three side rectangles have areas , and . Find the total surface area.

Potential Misunderstandings

• Students may confuse a prism with any three-dimensional solid
• Students may think a net is a side view rather than all faces laid flat
• Students may confuse total surface area with volume
• Students may Use linear units or cubic units instead of square units
• Students may forget to include all faces when finding TSA
• Students may count one or more faces twice, or miss hidden faces
• Students may add edge lengths instead of areas of faces
• Students may not Recognise that the two end faces of a prism are congruent
• Students may make errors finding the area of triangular end faces before adding all face areas

Next: 126. Volume and Capacity of Rectangular Prisms