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Maths · Transformations and constructions
3D solids
Name and describe 3D solids by their faces, edges and vertices, and draw and read plans and elevations.
Warm-up
Answer each one, then check.
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1
How many faces does a cube have?
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6
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2
What is the name of a 3D shape with a circular base and one curved surface ending at a point?
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A cone
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3
What is the volume of a cuboid 2 by 3 by 4?
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24
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4
What shape is the cross-section of a cylinder cut straight across?
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A circle
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5
What is a polygon with 6 sides called?
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A hexagon
Learning Objectives
- 1Name common 3D solids.
- 2Count faces, edges and vertices.
- 3Draw and interpret plans and elevations.
- 4Use a 3D drawing to work out how many cubes make a solid.
Faces, Edges and Vertices
A face is a flat surface, an edge is where two faces meet, and a vertex is a corner.
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Cuboid
Faces: 6. Edges: 12. Vertices: 8
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Triangular prism
Faces: 5. Edges: 9. Vertices: 6
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Square-based pyramid
Faces: 5. Edges: 8. Vertices: 5
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Cylinder
Faces: 3. Edges: 2. Vertices: 0
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Cone
Faces: 2. Edges: 1. Vertices: 1
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Sphere
Faces: 1. Edges: 0. Vertices: 0
Naming Solids
The name comes from the shape of the base or cross-section.
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Prism
The same cross-section all the way along: a triangular prism, a hexagonal prism.
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Pyramid
A polygon base with triangular faces that meet at one apex.
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Cylinder, cone, sphere
Solids with curved surfaces; a sphere is a perfect ball.
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Tetrahedron
A pyramid with four triangular faces.
EULER'S FORMULA
For any solid with flat faces, faces plus vertices minus edges equals 2.
Cuboid: \(6 + 8 - 12 = 2\). Square-based pyramid: \(5 + 5 - 8 = 2\).
Plan and Elevations
The plan is the view from above; elevations are views from the front and the side.
The Three Views
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Plan
The view from directly above.
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Front elevation
The view from the front, looking straight at the face you choose to call the front.
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Side elevation
The view from one side, usually the right, looking straight at that face.
Drawing the Elevations
A solid is made from four cubes: a row of three, with one more cube on top of the right-hand cube. Draw the front elevation, the plan and the side elevation viewed from the right.
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- 1 Front elevation: what you see facing the row Three squares wide; the right column is two high
- 2 Plan: look down from above Three squares in a row (the top cube is hidden above the right one)
- 3 Side elevation from the right: you see the end One square wide and two squares high
AnswerFront elevation: an L-shape of 4 squares. Plan: 3 squares in a row. Side elevation: 2 squares in a column.
Tips for Elevations
Draw only what you can see, with straight lines.
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Hidden edges
Do not draw edges you cannot see from that direction.
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Use a grid
One square on the grid stands for one edge of a cube.
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Check your views agree
The front elevation and plan must have the same width; the front and side must have the same height.
Build and Draw
Build a solid from 5 cubes: a row of three cubes, a fourth cube placed directly behind the middle cube of the row, and a fifth cube on top of the left-hand cube of the row. Draw its plan and its front elevation.
1. Build the solid.
2. Look from each direction.
3. Draw only what you see.
A good answer shows: Plan: a row of three squares with a fourth square behind the middle one (a T-shape). Front elevation: three squares wide, with the left column two squares high and the other two columns one square high.
Can I...?
- 1Name the common solids.
- 2Count faces, edges and vertices.
- 3Check with Euler's formula.
- 4Say what a plan is.
- 5Draw a front elevation.
- 6Draw a side elevation.
- 7Match views to a solid.
- 8Work out the volume of a cube solid from its views.
Summary & Exam Focus
- Faces, edges, vertices: count carefully and check with F + V − E = 2.
- Plan is from above; elevations are from the sides.
- Draw with straight lines on a grid.
- Compare your views for consistency.
Exam focus
A solid is made of 4 cubes: a row of three cubes with one cube on top of the right-hand cube. Draw the front elevation of the solid on a square grid. (2 marks) (2 marks)
Look at the solid from the front only and draw what is visible: an L-shape here. Do not add depth lines.
Key terms
The vocabulary this lesson expects you to use. Each one is linked from the first place it appears above.
- Face
- A flat surface of a solid.
- Edge
- A line where two faces meet.
- Vertex
- A corner where edges meet (plural: vertices).
- Plan
- The view of a solid from above.
- Elevation
- The view of a solid from the front or side.
- Prism
- A solid with the same cross-section all the way through.
Practice questions
Have a go at each one before you open its answer.
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Question 1 Non-calculator 2 marks
Write down the number of faces, edges and vertices of a square-based pyramid.
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Model answer
5 faces, 8 edges and 5 vertices.
Mark scheme
- Two correct — B1
- All three correct — B1
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Question 2 Non-calculator 2 marks
A solid has 5 faces, 9 edges and 6 vertices. Two of its faces are triangles. Write down the name of the solid.
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Model answer
A triangular prism.
Mark scheme
- Triangular prism — B2
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Question 3 Non-calculator 2 marks
Write down the shape of the plan of a cylinder standing on a flat table.
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Model answer
A circle.
Mark scheme
- Circle — B2
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Question 4 Non-calculator 2 marks
The diagram shows a solid made from four identical cubes. On the grid, draw the front elevation of the solid, viewed from the direction of the arrow F.
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Model answer
An L-shape made of 4 squares: a row of 3 squares with one more square above the right-hand square.
Mark scheme
- Correct number of columns and heights — M1
- A fully correct front elevation — A1
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Question 5 Non-calculator 2 marks
For the solid in the last question, draw the plan.
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Model answer
A row of 3 squares.
Mark scheme
- Three squares in a row — B2
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Question 6 Non-calculator 3 marks
For the same solid, draw the side elevation viewed from the right.
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Model answer
A column of 2 squares (one square wide, two squares high).
Mark scheme
- One square wide — M1
- Two squares high — M1
- A fully correct drawing — A1
Quick check
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How many vertices does a cuboid have?
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C: 8
A cuboid has 8 corners.
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What is a view of a solid from above called?
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B: Plan
The plan is the view from above.
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How many edges does a triangular prism have?
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D: 9
3 on each triangle plus 3 joining them: 9.
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A cone has how many flat faces?
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A: 1
One: the circular base.
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Which solid has 1 face, 0 edges and 0 vertices?
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C: Sphere
A sphere has one curved surface.
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Euler's formula says F + V − E equals...
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B: 2
For any solid with flat faces, it equals 2.
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