Shapes & nets
🎯 In this topic you will
- Identify, describe, and sketch compound three-dimensional shapes.
- Identify and sketch different nets for cubes, cuboids, prisms, and pyramids.
- Explain the relationship between the area of two-dimensional shapes and the surface area of three-dimensional shapes.
🧠 Key Words
- compound shape
- prism
- surface area
Show Definitions
- compound shape: A shape made by combining two or more simple shapes to form a more complex figure.
- prism: A three-dimensional shape with two identical parallel faces connected by rectangular or parallelogram sides.
- surface area: The total area of all the outer faces of a three-dimensional shape.
📦 Shapes of Parcel Boxes
When you see a person delivering a parcel to someone, what is the usual shape of the box? You will probably say a cuboid, but it is possible to have boxes that are cubes, pyramids or prisms as well. If you are working in a factory that makes the boxes, you need to know what to do to make the different shape boxes!
💡 Quick Math Tip
Split Compound Shapes: When a 3D shape looks complicated, try breaking it into simpler shapes such as cubes or cuboids. This makes it easier to describe the shape and work out measurements like surface area or volume.
❓ EXERCISES
1. Describe these compound shapes.
👀 Show answer
a. A cube and a square-based pyramid.
b. A cuboid and a triangular prism.
c. A cylinder and a cone.
2. Sort these shapes into two groups. Group $1$: simple shapes and group $2$: compound shapes.
👀 Show answer
Group $1$ (simple shapes): $A$, $D$, $E$
Group $2$ (compound shapes): $B$, $C$, $F$, $G$, $H$
3. Sketch a compound shape that is made from these simple shapes.
a. two different cuboids
b. a cuboid and a square-based pyramid
c. two different cylinders
👀 Show answer
Answers may vary. Each sketch should combine the given simple shapes to form one compound three-dimensional shape.
4. Use the same method as Deema to describe and sketch a net of these shapes.
a. cube
b. square-based pyramid
c. cylinder
d. triangular-based pyramid
👀 Show answer
a. A cube net has $6$ square faces arranged so they fold into a cube.
b. A square-based pyramid net has $1$ square base and $4$ triangular faces.
c. A cylinder net consists of $1$ rectangle and $2$ circles.
d. A triangular-based pyramid net has $4$ triangular faces.
🧠 Think like a Mathematician
a. Marcus asks this question:
How do you work out the surface area of a cuboid?
What do you think Marcus means by the surface area of a cuboid?
How do you think he could work it out?
b. How could you work out the surface area of:
i. a cube
ii. a square based pyramid
c. Copy and complete this general rule:
The surface area of a 3D shape is the total area of all its ________.
d. Think about your answers to parts a to c and explain your reasoning.
Show Answers
- a: Surface area means the total area of all the outer faces of a three-dimensional shape.
- b i: A cube has 6 equal square faces, so its surface area can be calculated by adding the areas of the six squares.
- b ii: A square-based pyramid has 1 square base and 4 triangular faces. Add the area of the base and the areas of the four triangles.
- c: The surface area of a 3D shape is the total area of all its faces.
- d: To calculate surface area, break the shape into its individual faces, find the area of each face, and add them together.
❓ EXERCISES
5. This diagram shows a triangular prism.
a. Copy and complete this description of the triangular prism.
A triangular prism has a total of ______ faces.
Two of the faces are ______ and ______ of the faces are rectangles.
b. Sketch a net for the triangular prism.
👀 Show answer
a. A triangular prism has a total of $5$ faces.
Two of the faces are triangles and $3$ of the faces are rectangles.
b. A correct net consists of $3$ rectangles joined in a strip with a triangle attached to each end.
6. Match each of these shapes to the correct net.
👀 Show answer
A. iii
B. i
C. ii
🧠 Think like a Mathematician
This shape is made of unit cubes.
a. What is the smallest number of unit cubes that must be added to the shape to make a cuboid?
b. Write down the method that you used to work out the answer to part $a$.
c. Think about your method and decide whether a different method could also work. Which method do you think is the best?
Show Answers
- a: The smallest number of unit cubes that must be added is $5$.
- b: First imagine the smallest cuboid that could fit around the whole shape. Then count how many cubes are already there and how many cubes are missing. The missing cubes are the cubes that must be added.
- c: Another method is to look at the shape layer by layer and fill in the gaps on each layer. The best method is usually the one that helps you see all the missing spaces clearly and avoids double counting.
💡 Quick Math Tip
Complete the Cuboid: Imagine the smallest cuboid that could contain the whole shape of cubes. Then look for the empty spaces inside that cuboid. The number of missing cubes shows how many cubes must be added to complete the cuboid.
❓ EXERCISES
7. Write down the smallest number of unit cubes that must be added to these shapes to make cuboids.
👀 Show answer
a. $2$ cubes
b. $3$ cubes
c. $4$ cubes
🧠 Think like a Mathematician
a. Choose a simple 3D shape and draw a net for that shape on a piece of paper.
b. Cut the net out, using a pair of scissors, and fold your net to make the shape.
c. What do you think of your net? Did it fold together accurately to make the shape or did some corners not meet? Did you have any faces missing, or faces that were the wrong shape?
d. Give yourself a score out of $10$ for your net, with $1$ being not very good and $10$ being perfect. How could you improve your score if you made the net again?
e. Think about your answers to parts $c$ and $d$ and explain how you could improve your method next time.
Show Answers
- a: A correct response could be drawing the net of a cube, cuboid, pyramid, or cylinder.
- b: When folded, the faces should meet correctly to form the three-dimensional shape.
- c: If the net was drawn correctly, the edges should meet neatly and no faces should overlap or be missing.
- d: Scores will vary. A score closer to $10$ means the net folds perfectly to form the shape.
- e: Improvement might include measuring more carefully, ensuring the faces are the correct shapes and sizes, and checking that all faces are connected properly before cutting.