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GCSE Maths · Edexcel 1MA1 · Foundation & Higher · Grades 2–5 · Geometry

3D Solids, Plans and Elevations

A plan is the view from above; an elevation is the view from the front or side. These questions are quick marks on Foundation papers. Imagine looking straight at the shape from one direction, and draw only what you would see.

1Faces, edges and vertices

A face is a flat (or curved) surface, an edge is where two faces meet, and a vertex is a corner.

SolidFacesEdgesVertices
Cube6128
Cuboid6128
Triangular prism596
Square-based pyramid585
Tetrahedron (triangular pyramid)464
Pentagonal prism71510
Cylinder3 (2 flat, 1 curved)20
Cone2 (1 flat, 1 curved)11
Sphere1 (curved)00
Euler’s formulaFor any polyhedron (flat faces only): F + V − E = 2. Use it to check your counts.

2Plans and elevations

  • Plan: the view from directly above.
  • Front elevation: the view from the front.
  • Side elevation: the view from the side.

Each view is drawn as a flat 2D shape. You don’t show depth, only the outline and any edges you can see.

Front
A shape made from 8 cubes. The front is the side facing bottom-left.
PlanFront elevationSide elevation
Reading the views
How do the views match the drawing?
  1. Plan: look down and you see the squares the cubes stand on. The back row has a gap in the middle.
  2. Front elevation: for each column, the tallest stack anywhere along it. The left column is 3 high, because of the back stack of 3.
  3. Side elevation (viewed from the right): for each row from front to back, the tallest stack in that row: 2 (front row) and 3 (back row). The front row appears on the left.
Each view shows the tallest stack in each line of sight
Exam tipUse squared or isometric paper and count squares carefully. On a plan, don’t show the heights, only which squares are covered.

3Plans and elevations of common solids

Cylinder: the plan is a circle and both elevations are rectangles.

PlanFrontSide

Square-based pyramid: the plan is a square with diagonals (the sloping edges) and both elevations are triangles.

PlanFrontSide

Cone: the plan is a circle with a dot at the apex, and the elevations are triangles.

PlanFrontSide

4Working back from the views

If you know the plan and elevations, you can work out the solid, and often its volume.

Worked example
A prism has a constant cross-section. Its front elevation is the shape below (each square is 1 cm) and its depth is 4 cm. Find its volume.
Front elevation
  1. The front elevation is the cross-section. Area = 5 squares = 5 cm².
  2. Volume of a prism = area of cross-section × depth = 5 × 4.
20 cm³
Common mistakes
  • Drawing a 3D sketch instead of a flat 2D view.
  • Including height information on a plan. A plan only shows which squares are covered.
  • Mixing up front and side. Check which direction the arrow or label marks as the front.
  • Counting a curved surface as several faces, or giving a cylinder vertices.

★Key Vocabulary Recap

Plan
View from directly above.
Elevation
View from the front or side.
Prism
A solid with the same cross-section all the way through.
Polyhedron
A 3D solid with flat faces.
Euler’s formula
F + V − E = 2 for any polyhedron.
Specification reference (Pearson Edexcel GCSE (9–1) Mathematics 1MA1: G12, G13)
  • G12: identify the properties of faces, surfaces, edges and vertices of cubes, cuboids, prisms, cylinders, pyramids, cones and spheres.
  • G13: construct and interpret plans and elevations of 3D shapes.
Next stepReady to practise? Try the 10 exam-style questions, then check your working against the worked solutions.
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Practice questions

GCSE Maths · Edexcel 1MA1 · Foundation & Higher · Grades 2–5 · Geometry

3D Solids, Plans and Elevations — Practice Questions

10 exam-style questions (19 marks), building from warm-up to grade 8–9. Show full working: Edexcel gives method marks even when the final answer is wrong.

Foundation · Grades 2–3
  1. (3 marks)
    Grade 2
    Write down the number of faces, edges and vertices of a triangular prism.
  2. (1 mark)
    Grade 2
    Name the solid that has six faces, all of which are squares.
  3. (1 mark)
    Grade 3
    How many cubes are used to make this shape? (Assume no cubes are hidden.)
    Front
  4. (2 marks)
    Grade 3
    On squared paper, draw the plan of the shape in Question 3.
Foundation and Higher · Grades 4–5
  1. (3 marks)
    Grade 4
    Draw the front elevation and the side elevation (from the right) of the shape in Question 3.
  2. (2 marks)
    Grade 4
    A cylinder stands on its circular base. Sketch its plan and its front elevation.
  3. (1 mark)
    Grade 4
    A solid has these views. Name the solid.
    PlanFrontSide
  4. (1 mark)
    Grade 4
    A solid has a plan that is a circle with a dot at its centre, and front and side elevations that are both triangles. Name the solid.
  5. (2 marks)
    Grade 5
    A polyhedron has 8 vertices and 12 edges. How many faces does it have?
  6. (3 marks)
    Grade 5
    A prism has this front elevation (each square is 1 cm by 1 cm). The prism is 4 cm deep. Work out its volume.
    Front elevation
When you’re doneRevise the method in the notes, then mark your answers with the worked solutions.
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Worked solutions

Show worked solutions
GCSE Maths · Edexcel 1MA1 · Foundation & Higher · Grades 2–5 · Geometry

3D Solids, Plans and Elevations — Worked Solutions

Full worked solutions to all 10 questions (19 marks), set out the way an Edexcel examiner wants to see them.

Foundation · Grades 2–3
  1. (3 marks)
    Grade 2
    Write down the number of faces, edges and vertices of a triangular prism.
    Solution
    1. Faces: 2 triangles + 3 rectangles = 5.
    2. Edges: 3 + 3 + 3 = 9.
    3. Vertices: 3 + 3 = 6. Check: 5 + 6 − 9 = 2 ✓
    5 faces, 9 edges, 6 vertices
  2. (1 mark)
    Grade 2
    Name the solid that has six faces, all of which are squares.
    Solution
    1. Six square faces → cube.
    Cube
  3. (1 mark)
    Grade 3
    How many cubes are used to make this shape? (Assume no cubes are hidden.)
    Front
    Solution
    1. Count the stacks: front row 1, 1, 2; middle row 1, 0, 3; back row 2, 0, 0.
    2. 1 + 1 + 2 + 1 + 3 + 2 = 10.
    10 cubes
  4. (2 marks)
    Grade 3
    On squared paper, draw the plan of the shape in Question 3.
    Solution
    1. Look down from above. Mark every square that has at least one cube on it.
    Plan
    See plan
Foundation and Higher · Grades 4–5
  1. (3 marks)
    Grade 4
    Draw the front elevation and the side elevation (from the right) of the shape in Question 3.
    Solution
    1. Front: tallest stack in each column, left to right: 2, 1, 3.
    2. Side (from the right): tallest stack in each row, front to back: 2, 3, 2.
    Front elevationSide elevation
    Front heights 2, 1, 3; side heights 2, 3, 2
  2. (2 marks)
    Grade 4
    A cylinder stands on its circular base. Sketch its plan and its front elevation.
    Solution
    PlanFrontSide
    Plan: circle. Front: rectangle
  3. (1 mark)
    Grade 4
    A solid has these views. Name the solid.
    PlanFrontSide
    Solution
    1. Square base seen from above, with edges meeting at the centre; triangles from the front and side.
    Square-based pyramid
  4. (1 mark)
    Grade 4
    A solid has a plan that is a circle with a dot at its centre, and front and side elevations that are both triangles. Name the solid.
    Solution
    PlanFrontSide
    Cone
  5. (2 marks)
    Grade 5
    A polyhedron has 8 vertices and 12 edges. How many faces does it have?
    Solution
    1. F + V − E = 2 → F + 8 − 12 = 2.
    2. F = 6.
    6 faces
  6. (3 marks)
    Grade 5
    A prism has this front elevation (each square is 1 cm by 1 cm). The prism is 4 cm deep. Work out its volume.
    Front elevation
    Solution
    1. Area of cross-section = 5 cm².
    2. Volume = 5 × 4 = 20.
    20 cm³
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