G12: 3D Shapes
Identify properties of cuboids, prisms, cylinders, pyramids, cones, spheres; faces, surfaces, edges, vertices
📋 Key Concepts
3D shapes have three dimensions: length, width, and height. They are solid objects that occupy space.
Key terms:
- Face: A flat surface of a 3D shape
- Edge: Where two faces meet
- Vertex (vertices): A corner point where edges meet
📝 Common 3D Shapes
| Shape | Faces | Edges | Vertices |
|---|---|---|---|
| Cube | 6 (squares) | 12 | 8 |
| Cuboid | 6 (rectangles) | 12 | 8 |
| Triangular prism | 5 | 9 | 6 |
| Square-based pyramid | 5 | 8 | 5 |
| Tetrahedron | 4 (triangles) | 6 | 4 |
| Cylinder | 3 (2 circles + curved) | 2 | 0 |
| Cone | 2 (circle + curved) | 1 | 1 |
| Sphere | 1 (curved) | 0 | 0 |
📝 Cubes and Cuboids
Cube: All 6 faces are squares. All edges are equal length. 12 edges, 8 vertices.
Cuboid: All 6 faces are rectangles. Opposite faces are equal. 12 edges, 8 vertices.
Example 1
A cuboid has dimensions 4 cm × 3 cm × 2 cm. Find the total length of all edges.
Solution:
A cuboid has 4 edges of each length.
Total = 4(4) + 4(3) + 4(2) = 16 + 12 + 8 = 36 cm
📝 Prisms
Prism: A 3D shape with the same cross-section throughout. The end faces are identical polygons, and the other faces are rectangles.
Types of prisms:
- Triangular prism
- Rectangular prism (cuboid)
- Pentagonal prism
- Hexagonal prism
Example 2
How many edges does a hexagonal prism have?
Solution:
Each hexagonal face has 6 edges.
There are 6 rectangular faces joining them.
Total edges = 6 + 6 + 6 = 18 edges
📝 Pyramids
Pyramid: A 3D shape with a polygon base and triangular faces meeting at a point (apex).
Square-based pyramid: 5 faces (1 square + 4 triangles), 8 edges, 5 vertices.
Tetrahedron (triangular-based pyramid): 4 triangular faces, 6 edges, 4 vertices.
📝 Cylinders, Cones and Spheres
Cylinder: 2 circular faces and 1 curved surface. 2 edges (the circles), no vertices.
Cone: 1 circular face and 1 curved surface. 1 edge (the circle), 1 vertex (the apex).
Sphere: A perfectly round 3D shape. 1 curved face, no edges, no vertices.
Example 3
Name a 3D shape with exactly one vertex.
Solution: A cone has exactly one vertex (the apex).
📝 Euler's Formula
Euler's Formula: For polyhedra (3D shapes with flat faces):
F + V - E = 2
Where F = number of faces, V = number of vertices, E = number of edges
Example 4
Verify Euler's formula for a cube.
Solution:
F = 6, V = 8, E = 12
F + V - E = 6 + 8 - 12 = 2 ✓
🧮 Worked Examples — Exam Style
3-D shape properties: count faces (flat surfaces), edges, vertices (corners). Euler: F + V − E = 2 for polyhedra.
Complete the table for a triangular prism: faces, edges, vertices.
Faces 5 (2 triangles + 3 rectangles), edges 9, vertices 6
B1 each. Check with Euler: 5 + 6 − 9 = 2 ✓ — the formula double-checks your counting.
Which 3-D shapes have NO vertices?
Sphere (no faces/edges/vertices) and cylinder (no vertices, though it has faces/edges of a curved kind)
B1 for sphere. Cylinders and cones are the boundary cases — cylinders have no vertices; cones have one apex (argued as a vertex).
Exam wording to expect: "How many faces/edges/vertices…", "name the solid", "verify Euler’s formula".
❓ Practice Questions
Q1: How many faces does a triangular prism have?
Q2: Name a 3D shape with no vertices.
Q3: How many edges does a square-based pyramid have?
Q4: What shape has 6 rectangular faces?
Q5: Verify Euler's formula for a tetrahedron.
Q6: Verify Euler's formula for a square-based pyramid
Q7: Which shape has 2 circular faces and 1 curved surface?
✅ Answers
- 5 faces (2 triangles + 3 rectangles)
- Sphere or cylinder
- 8 edges
- Cuboid (or rectangular prism)
- F = 4, V = 4, E = 6. F + V - E = 4 + 4 - 6 = 2 ✓
📋 Model Answers — Full Working
Every mark comes from the working, not just the answer. These model answers show the lines a mark scheme expects to see.
Q1. 5 faces (2 triangles + 3 rectangles)
Q2. Sphere (or cylinder — no corners)
Q3. 8 edges (4 base + 4 to the apex)
Q4. Cuboid (rectangular prism)
Q5. F = 4, V = 4, E = 6 → 4 + 4 − 6 = 2 ✓
Q6. F = 5, V = 5, E = 8 → 5 + 5 − 8 = 2 ✓
Q7. A cylinder
📌 Key Facts to Memorise
- Cube/cuboid: 6, 12, 8 · Triangular prism: 5, 9, 6 · Square pyramid: 5, 8, 5
- Tetrahedron: 4, 6, 4 · Octahedron: 8, 12, 6
- Euler’s formula: F + V − E = 2
- Prisms: two identical polygon ends + rectangles · Pyramids: one polygon base + triangles to an apex
- Curved solids (sphere, cylinder, cone) sit outside Euler’s rule
🗺️ In the Exam
3-D properties: 1–3 marks on Paper 1. Counting errors are common on hidden edges (pyramid base edges) — Euler’s check catches them.
Links to: plans/elevations (G13), volume (G16), nets.
🧭 Board Focus — OCR
Spec J560. Three papers of 100 marks each: Paper 1 non-calculator, Papers 2 and 3 calculator, at Foundation/Higher tiers. OCR marks problem-solving explicitly, and its papers tend to front-load multi-step worded contexts — practise extracting the mathematics from prose before calculating.
Always check: specification details change — confirm paper structure and set texts against the current specification for your exam year.
🎯 Exam Tips
- Learn the properties of each 3D shape
- Remember Euler's formula: F + V - E = 2
- Cylinders and cones have curved surfaces
- A sphere has no edges and no vertices
- Prisms are named by their cross-section shape
- Visualise unfolding 3D shapes into nets
🧠 Problem-Solving Strategies
Problem-Solving
Visualise 3D shapes by identifying faces, edges and vertices. Euler's formula: V - E + F = 2 for any convex polyhedron. For prisms: V = area of cross-section x length. For pyramids: V = 1/3 x base area x height. Use Pythagoras in 3D to find diagonal lengths by working through right-angled triangles.
Multi-Step Problem
A cuboid measures 4 cm by 3 cm by 12 cm. Find the length of its space diagonal (from one vertex to the opposite vertex).
Solution: First find the diagonal of the base: sqrt(4 squared + 3 squared) = sqrt(16 + 9) = 5 cm. Then the space diagonal = sqrt(5 squared + 12 squared) = sqrt(25 + 144) = sqrt(169) = 13 cm. Alternatively: sqrt(4 squared + 3 squared + 12 squared) = sqrt(16 + 9 + 144) = sqrt(169) = 13 cm.
⚠️ Common Errors
Watch Out!
1. Wrong: Calculating the volume of a pyramid as base area x height Correct: Volume of a pyramid = 1/3 x base area x height. The 1/3 is essential — a pyramid has one-third the volume of a prism with the same base and height.
2. Wrong: Forgetting that a cone is NOT a pyramid (different formula) Correct: A cone uses V = 1/3 x pi x r squared x h. A pyramid uses V = 1/3 x base area x h. The formulas are similar but the base shapes differ.
3. Wrong: Counting faces, edges or vertices incorrectly for complex polyhedra Correct: Count systematically. For a prism: F = n + 2, E = 3n, V = 2n where n = number of sides of the cross-section. Verify with Euler's formula: V - E + F = 2.
✍️ 6-Mark Exam Question
Extended Answer
6 marks: A solid is formed by placing a square-based pyramid on top of a cuboid. The cuboid has dimensions 6 cm x 6 cm x 8 cm. The pyramid has the same square base (6 cm x 6 cm) and a vertical height of 4 cm. (a) Find the volume of the combined solid. (b) Find the total surface area. (c) A sphere has the same volume as the combined solid. Find its radius to 1 decimal place.
(a) Cuboid volume = 6 x 6 x 8 = 288 cm cubed. Pyramid volume = 1/3 x 36 x 4 = 48 cm cubed. Total = 336 cm cubed.
(b) Cuboid SA (without top) = 4 x (6 x 8) + 6 x 6 = 192 + 36 = 228 cm squared. Pyramid: slant height = sqrt(9 + 16) = 5 cm. 4 triangular faces = 4 x 1/2 x 6 x 5 = 60 cm squared. Total SA = 228 + 60 = 288 cm squared.
(c) V = 4/3 x pi x r cubed = 336. r cubed = 336 x 3/(4 x pi) = 80.19. r = 4.3 cm (1 d.p.).
Mark scheme: M1 cuboid volume, A1 288, M1 pyramid volume, A1 total 336, M1 surface area components, A1 288, M1 sphere formula, A1 r = 4.3
📊 AO3: Reason & Interpret
Reasoning and Interpretation
A water tank is a cylinder with radius 30 cm and height 80 cm. Water is pumped in at 2 litres per minute.
(a) How long does it take to fill the tank? (1 litre = 1000 cm cubed)
(b) The tank is filled to a depth of 20 cm. A metal cube of side 15 cm is lowered into the water. By how much does the water level rise?
(c) A student says "A sphere with radius 30 cm would hold the same amount of water as this cylinder." Is this correct? Compare the volumes.
Answers: (a) Volume = pi x 30 squared x 80 = 72000pi cm cubed = 226,195 cm cubed = 226.2 litres. Time = 226.2/2 = 113.1 minutes (about 1 hour 53 minutes). (b) Cube volume = 15 cubed = 3375 cm cubed. Rise = 3375/(pi x 30 squared) = 3375/(900pi) = 1.19 cm. (c) Sphere volume = 4/3 x pi x 30 cubed = 36000pi. Cylinder = 72000pi. The sphere holds exactly HALF the cylinder. The student is wrong.
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