067. Nets and Platonic Solids

Learning Intentions

• To understand that a net is a two-dimensional representation of a solid’s faces
• To know what a polyhedron is
• To know what the five Platonic solids are
• draw a net of simple solids

Pre-requisite Summary

• Understand the difference between a 2D shape and a 3D solid
• Know that faces are the flat surfaces of a solid
• Be able to identify common faces such as squares, rectangles and triangles
• Understand that a solid can be unfolded into connected faces
• Know that not all solids are polyhedra, because some have curved surfaces
• Be able to sketch simple solids such as cubes, prisms and pyramids
• Understand that edges are where faces meet and vertices are where edges meet
• Be able to draw simple 2D shapes accurately with a ruler

Worked Examples

Worked Example 1

a) Explain what a net is.
b) Explain how a net is related to a 3D solid.
c) Decide whether a given arrangement of connected faces could be a net of a cube.

Worked Example 2

a) Define the term polyhedron.
b) Decide whether a cube is a polyhedron.
c) Decide whether a cylinder is a polyhedron and justify the answer.

Worked Example 3

Name the five Platonic solids:
a) list their names
b) state the shape of the faces of each
c) explain what they have in common

Worked Example 4

Draw a net of a cube:
a) draw six equal squares joined edge to edge
b) explain where the folds would occur
c) explain why all faces must be equal squares

Worked Example 5

Draw a net of a rectangular prism:
a) identify the six rectangular faces
b) arrange them so the solid can fold correctly
c) label matching faces

Worked Example 6

Draw a net of a triangular prism or square-based pyramid:
a) identify the faces needed
b) draw the faces joined edge to edge
c) explain how the net folds to make the solid

Problems

Problem 1

a) Explain what a net is.
b) Explain how a net is related to a 3D solid.
c) Decide whether a given arrangement of connected faces could be a net of a rectangular prism.

Problem 2

a) Define the term polyhedron.
b) Decide whether a triangular prism is a polyhedron.
c) Decide whether a cone is a polyhedron and justify the answer.

Problem 3

Name the five Platonic solids:
a) list their names
b) state the shape of the faces of each
c) explain what they have in common

Problem 4

Draw a net of a cube:
a) draw six equal squares joined edge to edge
b) explain where the folds would occur
c) explain why all faces must be equal squares

Problem 5

Draw a net of a rectangular prism:
a) identify the six rectangular faces
b) arrange them so the solid can fold correctly
c) label matching faces

Problem 6

Draw a net of a triangular prism or square-based pyramid:
a) identify the faces needed
b) draw the faces joined edge to edge
c) explain how the net folds to make the solid

Exercises

Understanding and Fluency

1. State whether each statement is true or false:
   a) A net is a flat drawing of the faces of a solid
   b) A sphere is a polyhedron
   c) A cube can have more than one possible net

2. State the meaning of each term:
   a) net
   b) polyhedron
   c) face

3. Decide whether each solid is a polyhedron:
   a) cube
   b) cylinder
   c) triangular prism

4. Decide whether each solid is a polyhedron:
   a) cone
   b) square-based pyramid
   c) rectangular prism

5. Name the five Platonic solids:
   a) solid with $4$ triangular faces
   b) solid with $6$ square faces
   c) solid with $8$ triangular faces

6. Name the five Platonic solids:
   a) solid with $12$ pentagonal faces
   b) solid with $20$ triangular faces
   c) solid with $6$ square faces

7. Draw a net for each solid:
   a) cube
   b) rectangular prism
   c) triangular prism

8. Draw a net for each solid:
   a) square-based pyramid
   b) triangular pyramid
   c) rectangular prism

9. For each solid, state the shapes needed in its net:
   a) cube
   b) triangular prism
   c) square-based pyramid

10. Mixed practice:
    a) Is a cone a polyhedron?
    b) Which Platonic solid has square faces?
    c) How many faces does a cube net need?

Reasoning

11. Explain why a cylinder is not a polyhedron.

12. A student says any arrangement of six squares is a cube net. Explain why this is incorrect.

13. Explain what all Platonic solids have in common.

14. A student says a net and a sketch of a solid are the same thing. Explain the difference.

15. Explain why the faces in a cube net must be connected edge to edge.

16. A student draws a net for a square-based pyramid using one square and three triangles. Explain the mistake.

Problem-solving

17. A packaging designer wants to make a cube-shaped box. Draw a possible net and state how many square faces are needed.

18. A classroom model is a triangular prism. Describe the faces needed and draw a possible net.

19. A museum display includes the five Platonic solids. List them and group them by the shape of their faces.

20. A student says a solid with curved surfaces can still be a Platonic solid. Explain why this is impossible.

21. A square-based pyramid is to be built from cardboard. Describe and draw a net that could be cut out and folded.

22. A solid has only flat faces, and every face is the same regular pentagon. Name the solid and explain how you know.

Potential Misunderstandings

• Students may think a net is just any flat picture of a solid rather than a foldable arrangement of its faces
• Students may confuse the faces of a solid with its edges or vertices
• Students may think any solid is a polyhedron, even if it has curved surfaces
• Students may confuse a polyhedron with a prism only
• Students may think there is only one possible net for a given solid such as a cube
• Students may forget that the faces in a net must be joined so they can fold without overlapping incorrectly
• Students may mix up the names of the Platonic solids
• Students may think Platonic solids only need equal numbers of faces, rather than congruent regular faces with the same arrangement at each vertex
• Students may omit a face or use the wrong face shapes when drawing a net of a solid