EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
3D solids
Transformations and constructions · Lesson 1 of 8
Teacher copy - includes the notes for whoever is teaching from it.
Warm-up
Answer each one, then check.
1. How many faces does a cube have?
6
2. What is the name of a 3D shape with a circular base and one curved surface ending at a point?
A cone
3. What is the volume of a cuboid 2 by 3 by 4?
24
4. What shape is the cross-section of a cylinder cut straight across?
A circle
5. What is a polygon with 6 sides called?
A hexagon
Learning Objectives
1. Name common 3D solids.
2. Count faces, edges and vertices.
3. Draw and interpret plans and elevations.
4. Use 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.
|
Solid |
Faces |
Edges |
Vertices |
|---|---|---|---|
|
Cuboid |
6 |
12 |
8 |
|
Triangular prism |
5 |
9 |
6 |
|
Square-based pyramid |
5 |
8 |
5 |
|
Cylinder |
3 |
2 |
0 |
|
Cone |
2 |
1 |
1 |
|
Sphere |
1 |
0 |
0 |
Naming Solids
The name comes from the shape of the base or cross-section.
▸ Prism. The same cross-section all the way along: a triangular prism, a hexagonal prism.
▸ Pyramid. A polygon base with triangular faces that meet at one apex.
▸ Cylinder, cone, sphere. Solids with curved surfaces; a sphere is a perfect ball.
▸ 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. |
An isometric solid of four cubes with its front elevation, plan and side elevation drawn on grids.
The Three Views
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Plan The view from directly above. |
Front elevation The view from the front, looking straight at the face you choose to call the front. |
|
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. |
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
Answer: Front 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.
▸ Hidden edges. Do not draw edges you cannot see from that direction.
▸ Use a grid. One square on the grid stands for one edge of a cube.
▸ Check your views agree. The front elevation and plan must have the same width; the front and side must have the same height.
Key Terms
|
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. |
Your Task: Build and Draw
12 minutes
|
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.
Note: Ask students to compare with a partner and check the three views agree.
Can I...?
☐ Name the common solids.
☐ Count faces, edges and vertices.
☐ Check with Euler's formula.
☐ Say what a plan is.
☐ Draw a front elevation.
☐ Draw a side elevation.
☐ Match views to a solid.
☐ Work out the volume of a cube solid from its views.
Summary
✓ 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.
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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) Look at the solid from the front only and draw what is visible: an L-shape here. Do not add depth lines. |
Exam Practice: 3D Solids
Answer all questions. · 20 minutes
▸ Question 1 · 2 marks · Non-calculator. Write down the number of faces, edges and vertices of a square-based pyramid.
▸ Question 2 · 2 marks · Non-calculator. A solid has 5 faces, 9 edges and 6 vertices. Two of its faces are triangles. Write down the name of the solid.
▸ Question 3 · 2 marks · Non-calculator. Write down the shape of the plan of a cylinder standing on a flat table.
▸ Question 4 · 2 marks · Non-calculator. 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…
▸ Question 5 · 2 marks · Non-calculator. For the solid in the last question, draw the plan.
▸ Question 6 · 3 marks · Non-calculator. For the same solid, draw the side elevation viewed from the right.
Question 1 · 2 marks · Non-calculator
|
“Write down the number of faces, edges and vertices of a square-based pyramid.” |
HOW TO ANSWER IT Command word: Non-calculator. Worth 2 marks, so plan before writing.
Question 1 · mark scheme
2 marks available. Award a mark for each point made.
▸ Two correct. B1
▸ All three correct. B1
▸ Model answer. 5 faces, 8 edges and 5 vertices.
Question 2 · 2 marks · Non-calculator
|
“A solid has 5 faces, 9 edges and 6 vertices. Two of its faces are triangles. Write down the name of the solid.” |
HOW TO ANSWER IT Command word: Non-calculator. Worth 2 marks, so plan before writing.
Question 2 · mark scheme
2 marks available. Award a mark for each point made.
▸ Triangular prism. B2
▸ Model answer. A triangular prism.
Question 3 · 2 marks · Non-calculator
|
“Write down the shape of the plan of a cylinder standing on a flat table.” |
HOW TO ANSWER IT Command word: Non-calculator. Worth 2 marks, so plan before writing.
Question 3 · mark scheme
2 marks available. Award a mark for each point made.
▸ Circle. B2
▸ Model answer. A circle.
Question 4 · 2 marks · Non-calculator
|
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. (2 marks) |
|
Question 4 · mark scheme
2 marks available. Award a mark for each point made.
▸ Correct number of columns and heights. M1
▸ A fully correct front elevation. A1
▸ Model answer. An L-shape made of 4 squares: a row of 3 squares with one more square above the right-hand square.
Question 5 · 2 marks · Non-calculator
|
“For the solid in the last question, draw the plan.” |
HOW TO ANSWER IT Command word: Non-calculator. Worth 2 marks, so plan before writing.
Question 5 · mark scheme
2 marks available. Award a mark for each point made.
▸ Three squares in a row. B2
▸ Model answer. A row of 3 squares.
Question 6 · 3 marks · Non-calculator
|
“For the same solid, draw the side elevation viewed from the right.” |
HOW TO ANSWER IT Command word: Non-calculator. Worth 3 marks, so plan before writing.
Question 6 · mark scheme
3 marks available. Award a mark for each point made.
▸ One square wide. M1
▸ Two squares high. M1
▸ A fully correct drawing. A1
▸ Model answer. A column of 2 squares (one square wide, two squares high).