3D Shapes
3D Shapes
- Unit 1: Angles & Constructions
- Unit 2: Shapes & Symmetry
- Unit 3: Position & Transformation
- Unit 4: Area, Volume & Symmetry
🎯 In this topic you will
- Identify and describe different 3D shapes
- Draw front, side, and top views of 3D shapes
- Find the connection between the number of vertices, faces, and edges of 3D shapes
- Draw accurate front, side, and top views of 3D shapes to scale
🧠 Key Words
- front view (front elevation)
- side view (side elevation)
- top view (plan view)
- visualise
Show Definitions
- front view (front elevation): The outline of a 3D object as seen directly from the front.
- side view (side elevation): The outline of a 3D object as seen directly from the side.
- top view (plan view): The outline of a 3D object as seen directly from above.
- visualise: To form a mental image of how a 3D shape looks from different perspectives.
3D Shapes You Should Know
You already know the names of some 3D shapes. Here are the 3D shapes you should know, as well as some 3D shapes that you might not have seen before.
Shapes shown: Cube, Cuboid, Tetrahedron (Triangular-based pyramid), Square-based pyramid, Sphere, Cone, Cylinder, Octahedron, Right-angled triangular prism, Equilateral triangular prism, Pentagonal prism, Trapezoidal prism.
You must be able to identify and describe the properties of a 3D shape. For example, a cuboid has:
- $6$ faces
- $8$ vertices
- $12$ edges
- all angles are right angles.
Tip
Remember that the vertices are the corners of a shape.
You must also be able to visualise and draw what a 3D shape looks like from different directions.
The top view is the view from above the shape.
The front view is the view from the front of the shape.
The side view is the view from the side of the shape.
Example: Look at this cylinder.
From above you will see a circle, but from the side and the front you will see a rectangle.
Tip
The top view can also be called the plan view or just plan. The side and front views can also be called the side and front elevations.
❓ EXERCISES
1. Match the properties (a to d) of each 3D shape with its name (A to D) and picture (i to iv). The first one has been done for you.
a. I have two faces that are congruent circles. I have one curved surface. I have two edges and no vertices.
👀 Show answer
b. I have four faces. All of my faces are congruent equilateral triangles. I have six edges and four vertices.
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c. I have only one curved surface. I have no edges or vertices.
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d. I have seven faces. Two of my faces are congruent pentagons and the other faces are congruent rectangles. I have 15 edges and ten vertices.
👀 Show answer
❓ EXERCISES
2. Write down the properties that characterise a cube.
You must include information about the faces, edges and vertices.
👀 Show answer
4. For each of the shapes given, work through this classification flow chart. Write down the letter you get at the end for each shape.
👀 Show answer
b. cone → $H$
c. triangular prism → $K$
d. sphere → $G$
e. octahedron → $L$
f. cylinder → $I$
❓ EXERCISES
5. Marcus makes a table that shows the number of faces, vertices and edges of different prisms. This is what he has done so far.
| Original shape | Number of sides | Shape of prism | Number of faces | Number of vertices | Number of edges |
|---|---|---|---|---|---|
| triangle | $3$ | triangular | $5$ | $6$ | $9$ |
| rectangle | $4$ | rectangular | $6$ | $8$ | $12$ |
| pentagon | $5$ | pentagonal | $7$ | $10$ | $15$ |
| hexagon | $6$ | hexagonal | $8$ | $12$ | $18$ |
| heptagon | $7$ | heptagonal | $9$ | $14$ | $21$ |
| octagon | $8$ | octagonal | $10$ | $16$ | $24$ |
a. Copy and complete the table.
👀 Show answer
b. Marcus says: “A triangle has three sides. A triangular prism has five faces, so the number of faces is two more than the number of sides of the original shape.” Is this true for every prism? Explain your answer.
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c. i Look back at the table. Compare the number of sides of the original shapes with the number of vertices of the prisms. What do you notice?
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c. ii Write down a general rule that connects the number of sides of the original shapes with the number of vertices of the prisms.
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d. i Look back at the table. Compare the number of sides of the original shapes with the number of edges of the prisms. What do you notice?
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d. ii Write down a general rule that connects the number of sides of the original shapes with the number of edges of the prisms.
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e. Use your answer to part d to complete this statement. The number of edges of a prism is always a multiple of ________.
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🧠 Think like a Mathematician
6. Work on your own to analyse the rules you found in parts $c$ and $d$ of Question 5.
Task:
- Explain clearly how each rule works using a labelled sketch of a general prism.
- Decide whether the rules are true for any prism and justify your decision.
- Write the rules as algebraic formulae.
Helpful approach: Let the original polygon have $n$ sides. Consider the two identical polygonal faces (top and bottom) and the $n$ rectangular side faces that join corresponding edges.
👀 show answer
- How the rules work: A prism has two copies of the base polygon (top and bottom). Each of the $n$ base vertices is paired with a matching top vertex, giving $2n$ vertices overall. Edges: there are $n$ around the bottom, $n$ around the top, and $n$ vertical joining edges → total $3n$. Faces: two polygonal faces (top and bottom) plus $n$ rectangular side faces → total $n+2$.
- Are the rules the same for any prism? Yes. The counts depend only on the base polygon having $n$ sides; they do not depend on the prism’s height or whether it is right or oblique. Therefore the relationships hold for every prism with an $n$-sided base.
- Algebraic formulae: Faces: $F = n + 2$ Vertices: $V = 2n$ Edges: $E = 3n$. (Check: $F + V - E = (n+2) + 2n - 3n = 2$, agreeing with Euler’s formula for polyhedra.)
❓ EXERCISES
7. The table shows the top view, front view and side view of some 3D shapes. The names of the 3D shapes are missing.
Write the missing names of the 3D shapes.
👀 Show answer
a. Rectangular prism (cuboid)
b. Pentagonal prism
c. Cone
d. Square-based pyramid
❓ EXERCISES
8. Draw the top view, front view and side view of a right-angled triangular prism.
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9. What 3D shapes have the same top view, front view and side view?
👀 Show answer
Sphere — all views are a circle.
Cube — all views are a square.
10. These diagrams show the top view of three prisms.
Match each diagram with the correct name of the prism given in the coloured box. Explain how you worked out your answer.
👀 Show answer
A →pentagonal prism
B →hexagonal prism
C →octagonal prism
Reasoning: From the top, a prism appears as a rectangle subdivided by visible side faces. The number of visible vertical strips increases with the number of sides of the base: A shows $2$ strips (so base has $5$ sides), B shows $3$ strips (base $6$ sides), C shows $4$ strips (base $8$ sides).
❓ EXERCISES
11. Emily is drawing 3D shapes on isometric paper. This is the first shape she draws.
a. Write down which of these is the front view, side view and top view: i, ii, iii.
b. Draw the front view, side view and top view for each of the shapes $i$ to $iv$. The arrows show the direction to look for front (F), side (S) and top (T).
👀 Show answer
a. Using the indicated viewing directions (F, S, T) on the isometric diagram:
- Front view:iii
- Side view:ii
- Top view:i
Reason (sketch-free): From the front you see a two-high column at one end with single cubes at the others (a “step” outline). From the side you see a straight row of equal-height unit squares. From the top you see three squares in a row with one extra square set back behind the middle square.
b. Describe each required orthographic view in words (you can sketch these from the descriptions):
Shape i (lower-left of the sheet):
- Top (T): a $2\times2$ square arrangement of unit squares (a filled $2$-by-$2$ footprint).
- Front (F): a rectangle of size $2$ wide by $2$ high (two squares across, two up).
- Side (S): a rectangle of size $2$ wide by $2$ high (identical to the front view).
Shape ii (upper-right among the four):
- Top (T): a $2\times2$ footprint.
- Front (F): a three-step outline: heights from left to right are $1$, then $2$, then $3$ unit squares (a vertical stack rising to the right).
- Side (S): two columns, heights $2$ and $3$ (rising toward the back, matching the F/S arrows).
Shape iii (lower-left among the four):
- Top (T): three squares in an “L” footprint (two in a row with one attached at the right-hand end to the back).
- Front (F): three columns with heights $1$, $1$, $2$ (a single cube at the left and centre, a two-high stack at the right).
- Side (S): two columns with heights $1$ and $2$ (the taller column at the back).
Shape iv (lower-right among the four):
- Top (T): an “L” footprint mirrored the other way (three in a row with one attached at the left-hand end to the back).
- Front (F): three columns with heights $1$, $2$, $3$ (rising to the centre column).
- Side (S): two columns with heights $1$ and $3$ (taller at the back).
How to check: For any isometric block model, the top view is the footprint (count squares seen from above); the front/side views are column-height profiles when viewed along the F or S arrow. Column heights are how many cubes lie in each vertical stack along that direction.
📐 Describing 3D Shapes
You already know how to describe a 3D shape using the number of faces, vertices and edges.
👁️ Views of 3D Shapes
You also know how to draw the top view, front view and side view of a 3D shape.
🔼 Top View
The top view is the view from above the shape.
It is sometimes called the plan view.
🏛️ Front View
The front view is the view from the front of the shape.
It is sometimes called the front elevation.
↔️ Side View
The side view is the view from the side of the shape.
It is sometimes called the side elevation.
📏 Drawing to Scale
You also need to be able to draw the top view, front view, and side view of a 3D shape to scale.
🧠 Think like a Mathematician
Task: Work independently to answer these questions about 3D shapes.
a. Copy and complete this table showing the number of faces, edges, and vertices of these 3D shapes.
| 3D shape | Number of faces | Number of vertices | Number of edges |
|---|---|---|---|
| cube | $6$ | $8$ | $12$ |
| cuboid | $6$ | $8$ | $12$ |
| tetrahedron | $4$ | $4$ | $6$ |
| square-based pyramid | $5$ | $5$ | $8$ |
| triangular prism | $5$ | $6$ | $9$ |
| trapezoidal prism | $6$ | $12$ | $18$ |
b. What is the connection between the number of faces, vertices and edges for all of the 3D shapes?
c. Write a formula that connects the number of faces (F), vertices (V) and edges (E).
d. Compare your formula with other possible versions. Do you have the same formula or a different one? Is your formula the same, just written in a different way?
e. Does your formula work for shapes with curved surfaces, or does it only work for shapes with flat faces? Explain your answer.
👀 Show Answers
- a: See completed table above.
- b: For all polyhedra, the relationship between faces, vertices, and edges follows Euler’s formula: $F + V = E + 2$.
- c: The formula is $F + V = E + 2$.
- d: Different versions may look like $E = F + V - 2$ or $V = E - F + 2$, but they are equivalent rearrangements.
- e: The formula only works for polyhedra with flat faces. It does not apply to shapes with curved surfaces like spheres or cylinders.
❓ EXERCISES
2. Copy and complete the workings and scale drawings for this question.
Draw the top view, front view, and side view of these shapes.
Use a scale of $1:2$.
| a. cube | b. cuboid | c. cylinder |
|---|---|---|
| Side length $6$ cm | Dimensions $8$ cm × $5$ cm × $3$ cm | Diameter $14$ cm, radius $7$ cm, height $10$ cm |
| $6 \div 2 =$ □ cm | $8 \div 2 =$ □ cm $3 \div 2 =$ □ cm $5 \div 2 =$ □ cm |
$7 \div 2 =$ □ cm $10 \div 2 =$ □ cm $14 \div 2 =$ □ cm |
a. Cube
💡 Tip
Draw a square of side length $3$ cm for the top, front, and side views.
b. Cuboid
💡 Tip
Top view: rectangle $4$ cm × $2.5$ cm.
Front view: rectangle □ cm × □ cm.
Side view: rectangle □ cm × □ cm.
c. Cylinder
💡 Tip
Top view: circle radius $3.5$ cm.
Front view: rectangle □ cm × □ cm.
Side view: rectangle □ cm × □ cm.
👀 Show answer
a. Cube: $6 \div 2 = 3$ cm. Top, front, and side views are all squares of side $3$ cm.
b. Cuboid: $8 \div 2 = 4$ cm, $3 \div 2 = 1.5$ cm, $5 \div 2 = 2.5$ cm.
Top view: rectangle $4$ cm × $2.5$ cm.
Front view: rectangle $4$ cm × $1.5$ cm.
Side view: rectangle $2.5$ cm × $1.5$ cm.
c. Cylinder: $7 \div 2 = 3.5$ cm, $10 \div 2 = 5$ cm, $14 \div 2 = 7$ cm.
Top view: circle radius $3.5$ cm.
Front view: rectangle $7$ cm × $5$ cm.
Side view: rectangle $7$ cm × $5$ cm.
🧠 Think like a Mathematician
3. Work independently to answer these questions.
Li and Seb are drawing the plan view of this shape.
This is what they draw.
a. Have either of them, or both of them, drawn the correct plan view?
b. Have they drawn their plan views to scale?
c. Reflect on your answers to parts a and b.
d. Draw the front elevation of the shape.
e. Draw the side elevation of the shape from
i. the left
ii. the right
Are your drawings for parts i and ii the same?
f. Reflect on and compare your drawings in parts d and e.
👀 show answer
- a:Seb has the correct plan view: a rectangle $3$ cm by $4$ cm with an internal vertical edge $1$ cm from the left (so widths $1$ cm and $2$ cm). Li shows only the outer outline, so his plan is incomplete.
- b: Yes. Both use the correct overall scale ($3$ cm × $4$ cm). Seb also places the internal edge to scale ($1$ cm from the left).
- c: Conclusion: Seb’s is correct and to scale; Li’s outline is to scale but missing the internal edge that separates the two heights.
- d: Front elevation (width = $3$ cm): a stepped outline — from left to right, height $5$ cm across the leftmost $1$ cm, then height $2$ cm across the remaining $2$ cm.
- e.i: Side elevation from the left: rectangle $4$ cm by $5$ cm (the taller side faces the left).
- e.ii: Side elevation from the right: rectangle $4$ cm by $2$ cm (the lower side faces the right). These drawings are not the same.
- f: Reflection: Differences arise because one side shows the taller $5$ cm section while the other shows the lower $2$ cm section. Ensure your elevations match these dimensions.
❓ EXERCISES
💡 Tip
If the views from the left side and from the right side are the same, you only need to draw one side elevation.
4. Draw the plan view, the front elevation and the side elevation of this 3D shape.
Use a scale of $1:10$.
👀 Show answer
Scale conversions ($1:10$):$60\text{ cm}\rightarrow 6\text{ cm}$, $25\text{ cm}\rightarrow 2.5\text{ cm}$, $35\text{ cm}\rightarrow 3.5\text{ cm}$, $20\text{ cm}\rightarrow 2\text{ cm}$, $15\text{ cm}\rightarrow 1.5\text{ cm}$, $30\text{ cm}\rightarrow 3\text{ cm}$.
Plan view: rectangle $6\text{ cm}\times 2.5\text{ cm}$.
Front elevation: step profile across the $6\text{ cm}$ width — left section height $3.5\text{ cm}$ for the first $3\text{ cm}$, then height $2\text{ cm}$ for the remaining $3\text{ cm}$.
Side elevation(s): from the left: rectangle $2.5\text{ cm}\times 3.5\text{ cm}$; from the right: rectangle $2.5\text{ cm}\times 2\text{ cm}$ (not the same, so draw both if required).
💡 Tip
A shipping container is a very large metal box used to move goods by lorry, train or ship.
5. The diagram shows the dimensions of a shipping container.
Ajani makes a house from three shipping containers.
The containers are arranged as shown in the diagram.
Draw the plan view, the front elevation, and the side elevation of his house.
Use a scale of $1:100$.
👀 Show answer
Scale conversions ($1:100$):$6\text{ m}\rightarrow 6\text{ cm}$, $2.6\text{ m}\rightarrow 2.6\text{ cm}$, $2.4\text{ m}\rightarrow 2.4\text{ cm}$, $3\text{ m}\rightarrow 3\text{ cm}$.
Plan view: three rectangles each $6\text{ cm}\times 2.4\text{ cm}$, stepped with gaps/offsets of $3\text{ cm}$ as shown.
Front elevation: faces of the containers are rectangles $2.4\text{ cm}\times 2.6\text{ cm}$; arrange them to match the stepping (containers behind may be hidden by the one in front).
Side elevation: rectangles $6\text{ cm}\times 2.6\text{ cm}$; show the three containers with their $3\text{ cm}$ offsets along the length.
🧠 Think like a Mathematician
6. Work independently to answer these questions.
The diagram shows two triangular prisms, A and B.
A is a right-angled triangular prism.
B is an isosceles triangular prism.
a. Will the side elevation of prism A be the same from the left side and the right side?
Explain your answer.
b. Draw the plan view, the front elevation, and the side elevation of prism A.
Use the actual dimensions shown.
Discuss the methods you could use to accurately draw the triangle. Which is the best method?
c. Will the side elevation of prism B be the same from the left side and the right side?
Explain your answer.
d. Draw the plan view, the front elevation, and the side elevation of prism B.
Use a scale of $1:4$.
Discuss the methods you could use to accurately draw the triangle. Which is the best method?
👀 show answer
- a:No. Prism A has a right-angled (scalene) triangular cross-section (sides $3$ cm, $4$ cm, $5$ cm). Left and right side elevations are mirror images, not identical, because the triangle has no line of symmetry.
- b:Views for A (full size):
- Plan: rectangle showing the prism length $6$ cm and triangle thickness (edge) as appropriate; outer outline is a $6$ cm by (triangle width) rectangle.
- Front elevation: right-angled triangle with legs $3$ cm (base) and $4$ cm (height), hypotenuse $5$ cm.
- Side elevation (left/right): show the triangular cross-section; left shows the right angle at the front edge, right is its mirror.
- Accurate triangle methods: (i) Mark a right angle with a set square, measure $3$ cm and $4$ cm legs, join to form the hypotenuse; (ii) Compass construction with circles radius $3$ cm and $4$ cm from adjacent vertices; (iii) Use a protractor to mark $90^\circ$ then measure sides. Best: set-square + ruler for guaranteed right angle and precise side lengths.
- c:Yes. Prism B is isosceles, so its triangular cross-section has a line of symmetry. Left and right side elevations are the same (not just mirrored).
- d:Views for B (scale $1:4$):
- Scaled lengths:$24\text{ cm}\rightarrow 6\text{ cm}$, $18\text{ cm}\rightarrow 4.5\text{ cm}$, $16\text{ cm}\rightarrow 4\text{ cm}$, $20\text{ cm}\rightarrow 5\text{ cm}$.
- Plan: rectangle of the scaled prism length (use the appropriate shown length, e.g., $6$ cm or $4.5$ cm depending on which edge represents the length) by the triangle thickness.
- Front elevation: isosceles triangle with equal sides scaled to $5$ cm and altitude $4$ cm.
- Side elevation: same on left and right (isosceles cross-section).
- Accurate triangle methods: (i) Draw the base to scaled length, construct a perpendicular bisector, mark the altitude $4$ cm to locate the apex; (ii) From each base endpoint draw arcs radius $5$ cm to find the apex; (iii) Use protractor with equal side lengths. Best: compass construction (equal radii) ensures true isosceles accuracy.
❓ EXERCISES
💡 Tip
Convert all the dimensions from metres to centimetres before using the scale to work out the dimensions of the scale drawings.
7. The diagram shows the dimensions of a village hall.
The roof is an isosceles triangular prism.
Draw the plan view, the front elevation, and the side elevation of the village hall.
Use a scale of $1:200$.
👀 Show answer
Convert to cm, then apply the scale $1:200$:
$12\text{ m}=1200\text{ cm}\rightarrow \dfrac{1200}{200}=6\text{ cm}$
$20\text{ m}=2000\text{ cm}\rightarrow \dfrac{2000}{200}=10\text{ cm}$
$4\text{ m}=400\text{ cm}\rightarrow \dfrac{400}{200}=2\text{ cm}$
$3\text{ m}=300\text{ cm}\rightarrow \dfrac{300}{200}=1.5\text{ cm}$
Plan view: rectangle $6\text{ cm}\times 10\text{ cm}$ (roof ridge not shown in plan for a prism roof).
Front elevation: rectangle $6\text{ cm}\times 2\text{ cm}$ with an isosceles triangle of height $1.5\text{ cm}$ centred on top (total height $3.5\text{ cm}$).
Side elevation: rectangle $10\text{ cm}\times 2\text{ cm}$ with an isosceles triangle of height $1.5\text{ cm}$ on top (roof ridge along the length).
❓ EXERCISES
8. The diagram shows a shape drawn on dotty paper.
The shape is made from $1$ cm cubes.
This diagram shows the plan view, front elevation and side elevation for the shape.
The diagrams have been drawn accurately on $1$ cm squared paper.
a. Which diagram, A, B or C shows the
i. plan view ii. front elevation iii. side elevation?
b. Is it possible to have a shape made from a different number of $1$ cm cubes which has the same plan view as the shape above? Explain your answer.
c. Is it possible to have a shape made from a different number of $1$ cm cubes which has the same plan view, front elevation, and side elevation as the shape above? Explain your answer.
👀 Show answer
a. Not enough information provided here to determine which of A, B, or C corresponds to each view without the precise cell layouts visible. Use the rule: plan = footprint (highest cube in each column), front = heights seen from the arrow labelled “Front”, side = heights from the arrow labelled “Side”. Match counts/step patterns to select A, B, C.
b.Yes. The plan view records only which ground positions are occupied by at least one cube, not how many cubes are stacked. You can change the number of cubes by increasing/decreasing stack heights while keeping the same footprint.
c.Yes, but only if the new shape preserves all three silhouettes. To have the same plan, front and side elevations, every column height must match when seen from both directions. Different internal arrangements (e.g., permuting cubes within columns) can give the same three views while using a different total number of cubes if some stacks are re-distributed but keep the same heights in both projections. If any column height changes, one of the elevations would differ.
9. The diagram shows four shapes drawn on dotty paper.
The shapes are made from $1$ cm cubes.
Draw accurately the plan view, front elevation and side elevation for each of the shapes.
Use $1$ cm squared paper.
The arrows in part a show the directions from which you should look at the shapes for the plan view (P), front elevation (F) and side elevation (S).
👀 Show answer
Method (for each shape a–d):
- Plan: From arrow P, mark the footprint squares occupied by at least one cube; shade one square per occupied column.
- Front: From arrow F, for each footprint column across the width, draw a stack whose height equals the number of cubes in that column.
- Side: From arrow S, repeat step 2 but across the depth direction.
- Use a scale of $1$ square = $1$ cm in each elevation; align stacks on grid lines.
Note: Multiple distinct 3D arrangements can produce identical pairs of elevations; check consistency between all three before finalising.
❓ EXERCISES
10. Accurately draw the outline of the plan view, front elevation and side elevation of this shape. Do not include any internal lines.
Question: Which of Marcus’s drawings is incorrect: the plan view, front elevation or side elevation? Explain the mistake he has made.
👀 Show answer
❓ EXERCISES
11. The diagram shows a shape drawn on dotty paper. The shape is made from cubes. The measurements of the shape are shown in the diagram. Accurately draw the plan view, front elevation, and side elevation for this shape. Use a scale of $1:2$ and use $1\ \text{cm}$ squared paper.
👀 Show answer
12. The diagram shows a shape drawn on dotty paper. The shape is made from cubes. The width of the shape is shown in the diagram. Accurately draw the plan view, front elevation, and side elevation for this shape. Use a scale of $1:3$ and use $1\ \text{cm}$ squared paper.
