125e. Nets and Surface Area of Prisms

Learning Intentions

• To know the meaning of the terms prism, net and total surface area (TSA)
• find 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 ${\text{cm}}^2$ and ${\text{m}}^2$ 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 $8\text{ cm}$, $5\text{ cm}$ and $3\text{ cm}$.

Worked Example 4

Find the total surface area of a cube with side length $6\text{ cm}$.

Worked Example 5

Find the total surface area of a triangular prism with:

a) triangular ends of base $4\text{ cm}$ and height $3\text{ cm}$
b) prism length $10\text{ cm}$
c) the other two sides of the triangular base both $5\text{ cm}$

Worked Example 6

A prism has total surface area made from:

a) two congruent end faces of area $12{\text{ cm}}^2$ each
b) three rectangular side faces of areas $30{\text{ cm}}^2$, $40{\text{ cm}}^2$ and $50{\text{ cm}}^2$
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 $9\text{ cm}$, $4\text{ cm}$ and $2\text{ cm}$.

Problem 4

Find the total surface area of a cube with side length $7\text{ cm}$.

Problem 5

Find the total surface area of a triangular prism with:

a) triangular ends of base $6\text{ cm}$ and height $4\text{ cm}$
b) prism length $8\text{ cm}$
c) the other two sides of the triangular base both $5\text{ cm}$

Problem 6

A prism has total surface area made from:

a) two congruent end faces of area $15{\text{ cm}}^2$ each
b) four rectangular side faces of areas $24{\text{ cm}}^2$, $24{\text{ cm}}^2$, $18{\text{ cm}}^2$ and $18{\text{ cm}}^2$
Find the TSA.

Exercises

Understanding and Fluency

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

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

3. Find the total surface area of each rectangular prism:
   a) $5\text{ cm}\times 4\text{ cm}\times 3\text{ cm}$
   b) $10\text{ cm}\times 6\text{ cm}\times 2\text{ cm}$
   c) $7\text{ m}\times 3\text{ m}\times 2\text{ m}$

4. Find the total surface area of each cube:
   a) side length $4\text{ cm}$
   b) side length $9\text{ cm}$
   c) side length $2.5\text{ m}$

5. Find the area of each net, then state the TSA of the prism:
   a) two ends of area $8{\text{ cm}}^2$ each, and side faces of area $12{\text{ cm}}^2$, $12{\text{ cm}}^2$, $20{\text{ cm}}^2$
   b) two ends of area $15{\text{ cm}}^2$ each, and side faces of area $18{\text{ cm}}^2$, $24{\text{ cm}}^2$, $30{\text{ cm}}^2$

6. Find the total surface area of each triangular prism:
   a) triangular end area $=10{\text{ cm}}^2$, and side rectangles of areas $15{\text{ cm}}^2$, $20{\text{ cm}}^2$, $25{\text{ cm}}^2$
   b) triangular end area $=12{\text{ cm}}^2$, and side rectangles of areas $18{\text{ cm}}^2$, $18{\text{ cm}}^2$, $30{\text{ cm}}^2$

7. A prism has the following face areas. Find the TSA:
   a) $6{\text{ cm}}^2,6{\text{ cm}}^2,14{\text{ cm}}^2,14{\text{ cm}}^2,21{\text{ cm}}^2,21{\text{ cm}}^2$
   b) $9{\text{ m}}^2,9{\text{ m}}^2,12{\text{ m}}^2,12{\text{ m}}^2,15{\text{ m}}^2,15{\text{ m}}^2$

8. Solve each:
   a) A rectangular prism has length $12\text{ cm}$, width $5\text{ cm}$ and height $4\text{ cm}$. Find its TSA
   b) A cube has TSA $150{\text{ cm}}^2$. Find the area of one face
   c) A prism has two end faces of area $11{\text{ cm}}^2$ and lateral faces totalling $42{\text{ cm}}^2$. Find the TSA

Reasoning

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

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

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

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

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

Problem-solving

14. A shoebox is a rectangular prism with dimensions $30\text{ cm}$, $18\text{ cm}$ and $12\text{ cm}$. Find its total surface area.

15. A cube-shaped gift box has side length $8\text{ cm}$. Find its total surface area.

16. A triangular prism has two triangular ends of area $9{\text{ cm}}^2$ each, and three rectangular side faces of areas $20{\text{ cm}}^2$, $24{\text{ cm}}^2$ and $28{\text{ cm}}^2$. Find the TSA.

17. A prism net has face areas $16{\text{ cm}}^2,16{\text{ cm}}^2,30{\text{ cm}}^2,30{\text{ cm}}^2,24{\text{ cm}}^2,24{\text{ cm}}^2$. Find the total surface area.

18. A rectangular water tank has length $2\text{ m}$, width $1.5\text{ m}$ and height $1\text{ m}$. Find its TSA.

19. A triangular prism has triangular ends each with area $14{\text{ cm}}^2$, and the three side rectangles have areas $35{\text{ cm}}^2$, $42{\text{ cm}}^2$ and $49{\text{ cm}}^2$. 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