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 questions.

What Are Faces, Edges, and Vertices?

• A face is a flat or curved surface on a 3D shape.
• An edge is a line segment where two faces meet.
• A vertex (plural: vertices) is a point where edges meet — a corner of the shape.

Euler's Formula

This formula works for all convex polyhedra (3D shapes with flat faces and no indentations). It does not apply to shapes with curved surfaces like cylinders, cones, or spheres.

Key 3D Shapes

Step-by-Step Method

Counting Faces, Edges, and Vertices

1. Identify the base shape — this tells you the type of prism or pyramid.
2. Count the faces: for a prism, it is the number of sides of the base + 2; for a pyramid, it is the number of sides of the base + 1.
3. Count edges and vertices systematically — base edges, top edges (prism) or apex edges (pyramid).
4. Verify with Euler's formula: F + V − E should equal 2.

Describing Cross-Sections

1. Identify the 3D shape and the direction of the cut (horizontal, vertical, or diagonal).
2. A horizontal cut through a prism parallel to the base gives a cross-section identical to the base.
3. A horizontal cut through a pyramid gives a smaller, similar version of the base.
4. Diagonal or vertical cuts produce different shapes depending on the angle.

Worked Example 1 — Foundation Level

Question: A shape has 6 faces and 8 vertices. Use Euler's formula to find the number of edges. Name the shape.

Working:

Step 1 — Euler's formula: F + V − E = 2.

Step 2 — 6 + 8 − E = 2, so 14 − E = 2, giving E = 12.

Step 3 — A shape with 6 faces, 12 edges, and 8 vertices is a cuboid (or cube).

Answer: The shape has 12 edges and is a cuboid.

Worked Example 2 — Higher Level

Question: A prism has a regular octagonal cross-section. Find the number of faces, edges, and vertices.

Working:

Step 1 — An octagonal prism has an 8-sided base and an 8-sided top, plus 8 rectangular lateral faces. Total faces = 8 + 2 = 10.

Step 2 — Edges: 8 on the base + 8 on the top + 8 vertical edges = 24.

Step 3 — Vertices: 8 on the base + 8 on the top = 16.

Step 4 — Check: F + V − E = 10 + 16 − 24 = 2. Correct.

Answer: 10 faces, 24 edges, and 16 vertices.

Worked Example 3 — Exam Style

Question: A horizontal cross-section is taken halfway up a square-based pyramid. Describe the shape of the cross-section and explain how it compares to the base.

Working:

Step 1 — A horizontal slice through a pyramid parallel to the base produces a shape similar to the base.

Step 2 — The base is a square, so the cross-section is also a square.

Step 3 — Because the cut is halfway up, the cross-section is smaller than the base. By similar shapes, the side length of the cross-section is half the base side length.

Answer: The cross-section is a square with side length half that of the base.

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.

Practice Questions

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.

Practise 3D shape questions with instant feedback free on GCSEMathsAI.

Related Topics

• Nets of 3D Shapes — unfolding 3D shapes into flat nets.
• Volume of 3D Shapes — calculating volumes of prisms, pyramids, and more.
• Plans and Elevations — viewing 3D shapes from different directions.

Summary

3D shapes questions test your knowledge of faces, edges, and vertices alongside naming and classifying solids. Euler's formula F + V − E = 2 is a powerful checking tool for polyhedra. For prisms, the cross-section is uniform and the number of lateral faces equals the number of sides of the base. For pyramids, all lateral faces are triangles meeting at an apex. Cross-section questions require you to visualise slicing through a solid — the shape you get depends on the direction and position of the cut. Always sketch, count systematically, and verify with Euler's formula.