15.1 3D Solids

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: naming solids, counting faces, edges and vertices, and drawing cuboids on isometric paper. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with interactive activities, practice quizzes and printable worksheets, in our free online course.

What you will learn

• Name common 3D solids such as prisms, pyramids, cylinders, cones and spheres
• Count the faces, edges and vertices of a solid
• Draw cuboids and shapes made from cubes on isometric paper
• Count the cubes in a solid drawn on isometric paper

Key words

• Face: A flat surface of a solid. A cylinder also has a curved surface.
• Edge: A line where two faces meet.
• Vertex (vertices): A corner where edges meet.
• Prism: A solid with the same cross-section all the way through, named after its end face.
• Pyramid: A solid with a polygon base and triangular faces that meet at one point (the apex).
• Isometric paper: Paper with dots arranged in triangles, used to draw 3D shapes.

Naming 3D solids

A 3D solid has length, width and height. You need to know these names and recognise them from a sketch. Dashed lines show edges hidden at the back.

A prism is named after its cross-section: a triangular prism has a triangle at each end, a hexagonal prism has a hexagon. A cube and a cuboid are special prisms. A pyramid is named after its base: a square-based pyramid has a square base.

Worked example: naming from a description

A solid has 2 hexagon faces and 6 rectangle faces. Name it.
Two identical end faces joined by rectangles means a prism. The ends are hexagons.

Answer: a hexagonal prism

Faces, edges and vertices

Count carefully and in an organised way: front face, back face, then the faces going round. For edges, count the front, the back, then the edges joining them.

Worked example: pentagonal prism

Faces: 2 pentagon ends + 5 rectangles round the side = 7
Edges: 5 on the front + 5 on the back + 5 joining them = 15
Vertices: 5 on the front + 5 on the back = 10

Answer: 7 faces, 15 edges, 10 vertices

• Cube or cuboid; Faces 6; Edges 12; Vertices 8
• Triangular prism; Faces 5; Edges 9; Vertices 6
• Square-based pyramid; Faces 5; Edges 8; Vertices 5
• Tetrahedron (triangle-based pyramid); Faces 4; Edges 6; Vertices 4
• Prism with an n-sided end; Faces n + 2; Edges 3n; Vertices 2n

Key fact: Check your counts: for any of these solids, faces + vertices − edges = 2. For the pentagonal prism, 7 + 10 − 15 = 2.

Drawing cuboids on isometric paper

Turn isometric paper so the dots make vertical columns. Vertical edges stay vertical. The other edges go along the sloping lines of dots. Each gap between two dots is 1 unit.

Worked example: drawing a 4 cm by 2 cm by 3 cm cuboid

1. Draw the front vertical edge 3 dots high.
2. From the bottom, draw 4 units up to the right and 2 units up to the left.
3. Draw the other vertical edges 3 units high, then join the top edges parallel to the bottom ones.
Hidden edges at the back are not drawn (or are dashed).

Answer: a cuboid with edges 4, 2 and 3 units long

Solids made from cubes

Shapes made from centimetre cubes are often drawn on isometric paper. Count layer by layer, and remember that a cube may be hidden behind or under others. In exam questions you are told if there are no hidden gaps.

Worked example: counting cubes

Bottom layer: 3 rows of 2 cubes = 6 cubes.
Top layer: 2 cubes standing on the left-hand end.

Answer: 6 + 2 = 8 cubes, so the volume is 8 cm³

Worked example: how many more?

How many more cubes are needed to turn the solid into a 3 by 2 by 2 cuboid?
A 3 by 2 by 2 cuboid has 3 × 2 × 2 = 12 cubes.

Answer: 12 − 8 = 4 more cubes

Common mistakes

• Counting only the edges you can see. A cube has 12 edges, even though a drawing shows fewer.
• Mixing up a prism and a pyramid. A prism has the same shape all the way through; a pyramid comes to a point.
• Drawing horizontal lines on isometric paper. Only vertical edges are drawn straight up; all other edges follow the sloping dots.
• Forgetting cubes hidden underneath other cubes when counting layers.

Revision questions

Try these, then check your answers below each one.

Question 1. How many vertices does a triangular prism have?

Answer: 6

Working: Each triangular end has 3 vertices: 3 + 3 = 6.

Question 2. How many edges does a square-based pyramid have?

Answer: 8

Working: 4 edges round the square base + 4 sloping edges = 8.

Question 3. Which solid has exactly one flat face and one curved surface? (Choose from: Cylinder, Cone, Sphere, Cuboid.)

Answer: Cone

Working: A cone has one flat circular base and one curved surface. A cylinder has two flat faces; a sphere has none.

Related lessons

• Previous lesson: 14.5 Direct and Inverse Proportion
• Next lesson: 15.2 Plans and Elevations
• In this chapter: 15.3 Accurate Drawings 1
• In this chapter: 15.4 Scale Drawings and Maps
• In this chapter: 15.5 Accurate Drawings 2

Continue in the full course

This lesson is section 15.1 (EGCMF1501) of our free GCSE Maths Foundation course. In the full course you get every video for this section, an animated interactive lesson, a self-marking practice quiz, a printable worksheet with a mark scheme, and your progress saved. Go to the full GCSE Maths Foundation course.