Sheet № 211 · Foundation + Higher · AQA · Edexcel · OCR
3D Shapes: Faces, Edges and Vertices –
Knowing the properties of 3D shapes — faces, edges, and vertices — is essential for GCSE Maths at both tiers. Exam questions ask you to name solids, count their features, apply Euler's formula, and describe cross-sections. This guide covers every 3D shape you need, explains the key relationships, and provides worked examples and practice
§Key definitions
Question:
A shape has 6 faces and 8 vertices. Use Euler's formula to find the number of edges. Name the shape.
Answer:
The shape has 12 edges and is a cuboid.
Q1 (Foundation):
A triangular prism has how many faces, edges, and vertices?
Q2 (Foundation):
Name a 3D shape with 5 faces, 8 edges, and 5 vertices.
Q3 (Higher):
A prism has 20 vertices. How many sides does the cross-section have? Find the number of faces and edges.
§Formulas to memorise
F + V − E = 2, where F = number of faces, V = number of vertices, E = number of edges
Verify with Euler's formula: F + V − E should equal 2.
Worked example
A shape has 6 faces and 8 vertices. Use Euler's formula to find the number of edges. Name the shape.
Working:
⚠ Common mistakes
- ✗Applying Euler's formula to curved shapes. Cylinders, cones, and spheres have curved faces, so F + V − E = 2 does not hold in the usual sense.
- ✗Confusing prisms and pyramids. A prism has two identical parallel bases and rectangular lateral faces. A pyramid has one base and triangular faces meeting at an apex.
- ✗Miscounting edges where faces meet. Count base edges and vertical/lateral edges separately, then add them together.
✦ Exam tips
- →Always verify your count with Euler's formula — if F + V − E does not equal 2 for a polyhedron, recount.
- →Know the vocabulary: a prism has a uniform cross-section; a pyramid tapers to a point.
- →Cross-section questions at Higher may involve cones (giving circles or ellipses) or cylinders (giving rectangles when cut vertically).
- →Sketch the shape if one is not provided — even a rough drawing helps you count features accurately.