Every 3D shape has three key features: faces (flat surfaces), edges (where two faces meet) and vertices (corners where edges meet). Euler's formula connects them all.
\[F + V - E = 2\]
Faces + Vertices − Edges = 2 (holds for all convex polyhedra)
Example 1 — Verify Euler's formula for a triangular prism
Count edges: 3 on each triangular end + 3 connecting edges = 9 edges
Step 3
Count vertices: 3 on each triangular end = 6 vertices
Step 4
Check: \(F + V - E = 5 + 6 - 9 = \mathbf{2}\) ✓
Example 2 — A shape has 6 faces and 8 vertices. How many edges does it have?
Step 1
Euler's formula: \(F + V - E = 2\)
Step 2
Substitute: \(6 + 8 - E = 2\)
Step 3
\(14 - E = 2\), so \(E = \mathbf{12}\) edges.
Memory tip: Vertices are the sharp corners — think of a room: it has 8 corners (vertices), 12 edges (where walls/ceiling/floor meet) and 6 faces (walls, floor, ceiling).
✓ Quick Check 1 — Faces, Edges, Vertices and Euler's Formula
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Prisms and Pyramids
Two major families of 3D shapes are prisms and pyramids. They look similar but have very different properties.
Prism: uniform cross-section throughout — two identical parallel end faces joined by rectangles
Pyramid: one polygonal base, all other faces are triangles meeting at a single apex
Example 3 — Identify properties of a pentagonal prism
Step 1
A pentagonal prism has a pentagon (5 sides) as its cross-section.
Prism vs Pyramid: A prism has TWO identical parallel bases. A pyramid has only ONE base. If you can slide the shape along and it looks the same throughout, it's a prism.
✓ Quick Check 2 — Prisms and Pyramids
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3D Shapes in Real Life and Cross-Sections
Recognising 3D shapes in everyday contexts and understanding their cross-sections is essential for functional maths.
Example 5 — Identify the cross-section of a cylinder and give a real-life example
Step 1
A cylinder has a circular cross-section — cutting it parallel to the ends always gives a circle.
Step 2
Real-life examples: a tin can, a drainpipe, a rolling pin, a toilet roll.
Example 6 — A builder cuts a triangular prism parallel to its end faces. What cross-section appears?
Step 1
A triangular prism has a uniform triangular cross-section.
Step 2
Cut parallel to the triangular ends → cross-section is a triangle.
Step 3
Cut parallel to a rectangular face → cross-section is a rectangle.
Exam tip: When asked for a cross-section, imagine slicing the shape like a loaf of bread. For any prism, the cross-section parallel to the ends is always the same shape as the end faces.
✓ Quick Check 3 — 3D Shapes in Real Life and Cross-Sections
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Practice
Work out each answer, then type it in and click Check to see if you're right.
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Question
Your answer
Result
1
How many faces does a cube have?
2
How many edges does a cuboid have?
3
How many vertices does a cube have?
4
A cylinder has 2 circular faces and 1 curved surface. How many faces does it have in total?
5
A triangular prism has F=5, E=9. Using Euler's formula F+V−E=2, how many vertices does it have?
6
How many faces does a square-based pyramid have?
7
A shape has 8 faces and 6 vertices. Using Euler's formula, how many edges does it have?
8
What is the cross-section shape of a cylinder when cut parallel to its circular ends?
9
How many edges does a triangular prism have?
10
A cone has 1 flat face and 1 curved surface. How many faces does it have in total?
⚠ Things to Watch Out For
Don't confuse faces, edges and vertices — a face is a flat surface, an edge is where two faces meet, and a vertex is a corner point.
Euler's formula states F + V − E = 2 — use it to check your counts, but miscount one and the whole check fails.
Cylinders and cones have curved surfaces, not flat faces — be careful when counting faces for these shapes.
A prism has two identical parallel end faces and rectangular side faces; a pyramid has one base and triangular faces meeting at an apex.
A sphere has no edges, no vertices, and just one curved surface — it does not follow Euler's formula.