8.4 Surface Area of 3D Solids

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: faces, edges and vertices, nets, and the surface area of cuboids and prisms. 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 3D solids and count their faces, edges and vertices
• Recognise and draw nets of 3D solids
• Work out the surface area of a cube and a cuboid
• Work out the surface area of a triangular prism

Key words

• Face: A flat surface of a solid.
• Edge: A line where two faces meet.
• Vertex: A corner, where edges meet. Plural: vertices.
• Net: A 2D shape that folds up to make a 3D solid.
• Prism: A solid with the same cross-section all the way through, such as a cuboid or a triangular prism.
• Surface area: The total area of all the faces of a solid. Units: cm², m², …

Faces, edges and vertices

• Cube / 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 (triangular-based pyramid); Faces 4; Edges 6; Vertices 4

Worked example: a pentagonal prism

A pentagonal prism has 2 pentagon ends and 5 rectangles: 7 faces.
Edges: 5 on each end + 5 joining the ends = 15. Vertices: 5 on each end = 10.

Answer: 7 faces, 15 edges, 10 vertices

Nets

Imagine cutting along some edges of a box and opening it out flat. The flat shape is a net. A solid can have several different nets.

Worked example: which solid?

A net has 1 square and 4 identical triangles, one on each side of the square. What solid does it make?
The triangles fold up and meet at a point above the square.

Answer: A square-based pyramid

Surface area of cubes and cuboids

A cuboid has three pairs of identical faces: top and bottom, front and back, two sides. Find one of each, add, then double.

Worked example: a cuboid

Front 6 × 3 = 18, top 6 × 4 = 24, side 4 × 3 = 12.
Surface area = 2 × (18 + 24 + 12) = 2 × 54 = 108

Answer: 108 cm²

Worked example: a cube

A cube has 6 identical square faces. Side 5 cm: 6 × 5 × 5 = 150

Answer: 150 cm²

Surface area of a triangular prism

A triangular prism has 2 triangle ends and 3 rectangles. Each rectangle is one side of the triangle × the length of the prism.

Worked example: a triangular prism

Triangles: 2 × (1/2 × 4 × 3) = 12 cm².
Rectangles: 4 × 10 + 3 × 10 + 5 × 10 = 40 + 30 + 50 = 120 cm².
Total = 12 + 120 = 132

Answer: 132 cm²

Key fact: Make a list of faces so you do not miss one: 2 triangles + 3 rectangles = 5 faces.

Common mistakes

• Finding only 3 faces of a cuboid and forgetting to double.
• Multiplying length × width × height – that is volume, not surface area.
• Using the same rectangle three times for a triangular prism when the triangle’s sides are different.
• Giving surface area in cm or cm³ instead of cm².

Revision questions

Try these, then check your answers below each one.

Question 1. How many edges does a cuboid have? (Choose from: 6, 8, 12, 24.)

Answer: 12

Working: 4 + 4 + 4 = 12 edges.

Question 2. Work out the surface area of a cuboid 4 cm by 3 cm by 2 cm.

Answer: 52 cm²

Working: 2 × (12 + 8 + 6) = 52 cm².

Question 3. Work out the surface area of a cube with edges 6 cm long.

Answer: 216 cm²

Working: 6 × 6 × 6 = 216 cm² (6 faces of 36 cm²).

Related lessons

• Previous lesson: 8.3 Area of Compound Shapes
• Next lesson: 8.5 Volume of Prisms
• In this chapter: 8.1 Rectangles, Parallelograms and Triangles
• In this chapter: 8.2 Trapezia and Changing Units
• In this chapter: 8.6 More Volume and Surface Area

Continue in the full course

This lesson is section 8.4 (EGCMF0804) 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.