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G12

Identify properties of the faces, surfaces, edges and vertices of: cubes, cuboids, prisms, cylinders, pyramids, cones and spheres

3D shape properties

Worked answers, methods and verified real exam appearances for G12 on Edexcel GCSE Maths 1MA1.

Explanation

  • Describe a solid by its flat faces, curved surfaces, edges and vertices. A face is flat; an edge is where surfaces meet; a vertex is where edges meet.
  • An nn-sided prism has two congruent polygonal ends and nn rectangular side faces, so it has n+2n+2 faces, 3n3n edges and 2n2n vertices.
  • A pyramid has one polygonal base and triangular faces meeting at an apex.
  • A cylinder has two circular faces and one curved surface but no vertices; a cone has one circular face, one curved surface and one vertex; a sphere has only one curved surface.
  • Count hidden features once.
A prism has congruent polygonal ends joined by rectangular faces.

Worked example

A prism has hexagonal ends. Work out its numbers of faces, edges and vertices.

  1. 1.For a prism with n=6n=6, faces =n+2=8=n+2=8.
  2. 2.Edges =3n=18=3n=18.
  3. 3.Vertices =2n=12=2n=12.

Answer: 88 faces, 1818 edges and 1212 vertices.

Common mistakes

  • Don't count a cylinder's curved surface as a flat face or give the cylinder vertices.
  • Don't count only visible edges and omit hidden edges at the back of a solid.

Exam tip

For a prism, identify the number of sides on one end first, then use n+2n+2, 3n3n and 2n2n as a check.

Worked practice

Q1
Tier 1 · Easy

1

Write down the number of faces and the number of vertices of a triangular prism.

(2)

(Total for Question 1 is 2 marks)

Mark scheme

Mark scheme for question 1
QuestionAnswerMarkMark scheme
1
  • 55 faces
  • 66 vertices
2A triangular prism has two triangular faces and three rectangular faces, making 55 faces. Its two triangular ends have 3+3=63+3=6 vertices.
Q2
Tier 2 · Standard

2

A solid has one circular plane face, one curved surface, one circular edge and one vertex. Name the solid.

(2)

(Total for Question 2 is 2 marks)

Mark scheme

Mark scheme for question 2
QuestionAnswerMarkMark scheme
2
  • Cone
2The circular base is the one plane face, its rim is the circular edge, the side is curved, and the surfaces meet at one apex. These are the properties of a cone.
Q3
Tier 3 · Hard

3

A prism has a regular 99-sided polygon as each end. Work out its numbers of faces, edges and vertices.

(3)

(Total for Question 3 is 3 marks)

Mark scheme

Mark scheme for question 3
QuestionAnswerMarkMark scheme
3
  • 1111 faces
  • 2727 edges
  • 1818 vertices
3For an nn-sided prism, the counts are n+2n+2 faces, 3n3n edges and 2n2n vertices. With n=9n=9, these are 1111 faces, 2727 edges and 1818 vertices.
Q4
Tier 1 · Easy

4

A cube and a cuboid both have 66 faces. Write down one other number that is the same for both and say what it counts.

(2)

(Total for Question 4 is 2 marks)

Mark scheme

Mark scheme for question 4
QuestionAnswerMarkMark scheme
4
  • 88 vertices (or 1212 edges)
2A cube and a cuboid each have 88 vertices and 1212 edges. Giving either number with the matching feature answers the question.
Q5
Tier 2 · Standard

5

Work out the numbers of faces, edges and vertices of a square-based pyramid.

(3)

(Total for Question 5 is 3 marks)

Mark scheme

Mark scheme for question 5
QuestionAnswerMarkMark scheme
5
  • 55 faces
  • 88 edges
  • 55 vertices
3There is one square base and four triangular faces, giving 55 faces. There are 44 base edges and 44 sloping edges, giving 88 edges. The 44 base vertices and the apex give 55 vertices.
Q6
Tier 3 · Hard

6

A prism has 1212 faces. Work out the number of sides on each congruent end, name the prism, and work out its numbers of edges and vertices.

(4)

(Total for Question 6 is 4 marks)

Mark scheme

Mark scheme for question 6
QuestionAnswerMarkMark scheme
6
  • 1010 sides; a decagonal prism; 3030 edges and 2020 vertices
4Subtract the two end faces from the total: 12−2=1012-2=10, so each congruent end is a decagon and the solid is a decagonal prism. Its edge count is 3×10=303\times10=30, and its vertex count is 2×10=202\times10=20.
Q7
Tier 2 · Standard

7

A solid has no vertices and two circular plane faces. Write down the name of the solid, its number of edges and its number of curved surfaces.

(3)

(Total for Question 7 is 3 marks)

Mark scheme

Mark scheme for question 7
QuestionAnswerMarkMark scheme
7
  • Cylinder; 22 edges; 11 curved surface
3The two circular plane faces and one curved surface identify a cylinder. Each circle where a plane face meets the curved surface is an edge, giving 22 edges, and the side is one continuous curved surface.
Q8
Tier 3 · Hard

8

One solid is a prism with 77-sided congruent ends. A second solid is a pyramid with a 77-sided base. Work out the total numbers of faces, edges and vertices across the two solids.

(4)

(Total for Question 8 is 4 marks)

Mark scheme

Mark scheme for question 8
QuestionAnswerMarkMark scheme
8
  • 1717 faces; 3535 edges; 2222 vertices
4The prism has 7+2=97+2=9 faces, 3×7=213\times7=21 edges and 2×7=142\times7=14 vertices. The pyramid has 7+1=87+1=8 faces, 7+7=147+7=14 edges and 7+1=87+1=8 vertices. The totals are 9+8=179+8=17 faces, 21+14=3521+14=35 edges and 14+8=2214+8=22 vertices.
Q9
Tier 3 · Hard

9

A pyramid has a polygon with nn sides as its base. The pyramid has 55 more edges than faces. Work out nn, name the pyramid, and write down its numbers of faces, edges and vertices.

(4)

(Total for Question 9 is 4 marks)

Mark scheme

Mark scheme for question 9
QuestionAnswerMarkMark scheme
9
  • n=6n=6; hexagonal-based pyramid; 77 faces, 1212 edges and 77 vertices
4An nn-sided pyramid has n+1n+1 faces and 2n2n edges. The condition gives 2n=(n+1)+52n=(n+1)+5, so n=6n=6. It is therefore a hexagonal-based pyramid. It has 6+1=76+1=7 faces, 2×6=122\times6=12 edges and 6+1=76+1=7 vertices.
Q10
Tier 3 · Hard

10

A triangular-based pyramid is joined to a triangular prism so that one triangular face of each solid matches exactly and is hidden inside the new solid. No other faces overlap, and no exterior faces become coplanar. Work out the numbers of exterior faces, distinct edges and distinct vertices of the new solid.

(5)

(Total for Question 10 is 5 marks)

Mark scheme

Mark scheme for question 10
QuestionAnswerMarkMark scheme
10
  • 77 exterior faces; 1212 distinct edges; 77 distinct vertices
5The triangular prism has 55 faces, 99 edges and 66 vertices. The triangular-based pyramid has 44 faces, 66 edges and 44 vertices. Joining the triangular faces removes two exterior faces, so there are 5+4−2=75+4-2=7 exterior faces. The three edges and three vertices around the join are shared, not doubled, giving 9+6−3=129+6-3=12 edges and 6+4−3=76+4-3=7 vertices. The check 7−12+7=27-12+7=2 confirms the counts.

Verified exam appearances

SeriesPaperQuestionMarksCalculatorTierLinks
2021-111FQ62Non-calculatorFoundationQPMS
2023-113FQ103AllowedFoundationQPMS
2022-063FQ92AllowedFoundationQPMS
2023-112FQ72AllowedFoundationQPMS
2023-063FQ82AllowedFoundationQPMS

Other points in G Geometry and measures

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