Lesson 12.2: Prisms, Pyramids, and Net Folding
Applied concepts and practical problem solving for Prisms, Pyramids, and Net Folding.
⏱️ Reading Time: 20 mins
• ❓ 50 Interactive Practice Questions
🌟
Real-Life Story: 2D Shapes and 3D Objects
Cardboard Box Factory: Die-cutting flat cardboard sheets into foldable cereal boxes.
💡 Core Mathematical Concepts: Prisms, Pyramids, and Net Folding
Focus areas: Faces, edges, vertices, and identifying valid foldable nets of 3D solids..
| Concept / Term | Mathematical Meaning | Real-Life Application |
|---|---|---|
| Prism | A 3D solid with two identical parallel end faces and rectangular sides | Triangular prism, rectangular prism (cuboid) |
| Pyramid | A 3D solid with a polygon base and triangular faces meeting at an apex | Square pyramid, triangular pyramid (tetrahedron) |
| 3D Net | A flat 2D pattern that can be folded along edges to form a 3D solid | Cereal box unfold pattern |
📝 Step-by-Step Worked Examples
Worked Example 1: Triangular Prism Properties
5 faces (2 triangular, 3 rectangular), 9 straight edges, 6 vertices.
Worked Example 2: Square Pyramid Properties
5 faces (1 square base, 4 triangular faces), 8 edges, 5 vertices.
⚠️ Common Misconceptions to Avoid
Common Misconception: Thinking a cylinder is a prism
Correction: Prisms must have flat polygon bases; cylinders have circular bases.
Common Misconception: Invalid net configurations
Correction: If faces overlap when folded, the 2D layout is not a valid net.
🎯
Self-Check Challenge
A triangular prism is placed on a table. How many rectangular faces and how many triangular faces does it have?
Answer & Explanation:
It has 3 rectangular side faces and 2 triangular end bases (5 faces total).
🎯 Key Learning Takeaways
Extend understanding of Prisms, Pyramids, and Net Folding.
Solve multi-step real-world problems systematically.
Verify solutions through mathematical reasoning.
📄 Printable Worksheets (5 Available)
Download ready-to-print PDFs for offline student practice with separate parent/teacher answer keys.
G5-M-CH12-WS05
A4 • 15 Qs
2D Shapes and 3D Objects: Worksheet 5
Practice Assessment 5 • Central Botanical Gardens Context
G5-M-CH12-WS04
A4 • 15 Qs
2D Shapes and 3D Objects: Worksheet 4
Practice Assessment 4 • Eco-Friendly Solar Park Context
G5-M-CH12-WS03
A4 • 15 Qs
2D Shapes and 3D Objects: Worksheet 3
Practice Assessment 3 • Harbor Marine Terminal Context
G5-M-CH12-WS02
A4 • 15 Qs
2D Shapes and 3D Objects: Worksheet 2
Practice Assessment 2 • Science & Discovery Museum Context
G5-M-CH12-WS01
A4 • 15 Qs
2D Shapes and 3D Objects: Worksheet 1
Practice Assessment 1 • City Regional Airport Context
✍️ Lesson Practice Questions (50 Questions)
Solve these real-world challenges online, test your understanding, and reveal instant step-by-step explanations.
1
Short Answer
Hard
2 Marks
A wooden cube has a side length of 4 cm. It is painted red on all outer surfaces and then cut into 1 cm unit cubes. How many 1 cm cubes have exactly 2 red faces?
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
24 cubes
Step-by-Step Explanation:
Cubes with 2 painted faces lie along the 12 edges (excluding corners). Each edge has 4 - 2 = 2 such cubes. 12 edges × 2 = 24 cubes.
2
Short Answer
Hard
2 Marks
Verify Euler's formula (F + V - E = 2) for a pentagonal prism. List its number of faces, vertices, and edges.
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
F = 7, V = 10, E = 15; 7 + 10 - 15 = 2
Step-by-Step Explanation:
Faces = 2 pentagons + 5 rectangles = 7. Vertices = 5 + 5 = 10. Edges = 5 + 5 + 5 = 15. 7 + 10 - 15 = 2.
3
Short Answer
Hard
2 Marks
Why are triangles considered the most rigid and structurally stable geometric shape in engineering bridge trusses?
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
A triangle's shape is completely locked by its side lengths. Unlike rectangles, which can deform into parallelograms under load, a triangle cannot change shape without bending or breaking a member.
Step-by-Step Explanation:
Triangulation provides geometric rigidity because side lengths uniquely fix internal angles.
4
Short Answer
Hard
2 Marks
Explain why a sphere cannot be formed by folding a flat sheet of 2D paper without cutting, stretching, or wrinkling.
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
A sphere has continuous double curvature across its surface, whereas a sheet of paper has zero curvature (flat plane). Flat paper cannot conform to spherical curvature without crumpling.
Step-by-Step Explanation:
Gaussian curvature of a sphere is positive, while paper is flat (zero curvature).
5
Short Answer
Medium
1 Mark
A student draws a net for a cube with 5 connected squares. Why cannot this net fold into a cube?
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
A cube must have 6 faces
Step-by-Step Explanation:
A cube requires exactly 6 square faces; a 5-square net leaves one side missing.
6
Short Answer
Medium
1 Mark
Lucas says: 'A cylinder has 2 edges and 2 vertices.' Explain Lucas's error and state the correct number of vertices on a cylinder.
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
A cylinder has 0 vertices
Step-by-Step Explanation:
The circular edges of a cylinder are smooth loops with no sharp corners (vertices). A cylinder has 0 vertices.
7
Fill in the Blanks
Medium
1 Mark
Euler's formula for polyhedra is Faces + Vertices - Edges = [ ? ].
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
2
Step-by-Step Explanation:
Euler's characteristic formula for convex polyhedra is F + V - E = 2.
8
Fill in the Blanks
Medium
1 Mark
Solid Shape Riddle: 'I have 6 identical flat square faces, 12 equal edges, and 8 corners.' What shape am I?
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
Cube
Step-by-Step Explanation:
A cube has 6 identical square faces, 12 edges, and 8 vertices.
9
Short Answer
Medium
1 Mark
A student glues together two identical wooden cubes face-to-face to form a single cuboid. How many flat faces does the new cuboid have?
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
6 faces
Step-by-Step Explanation:
A cuboid always has 6 flat rectangular faces.
10
Short Answer
Medium
1 Mark
A shipping crate in the shape of a rectangular cuboid has dimensions 5 m long, 3 m wide, and 2 m high. What is the total surface area of all 6 faces?
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
62 sq m
Step-by-Step Explanation:
2 × (5×3 + 5×2 + 3×2) = 2 × (15 + 10 + 6) = 2 × 31 = 62 square meters.
11
Short Answer
Medium
1 Mark
At the Central Botanical Gardens, an architect constructs a geometric installation containing 4 cubes and 3 triangular prisms. How many total flat faces exist across all 7 shapes?
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
39 faces
Step-by-Step Explanation:
4 cubes × 6 faces = 24. 3 triangular prisms × 5 faces = 15. Total = 24 + 15 = 39 flat faces.
12
Multiple Choice
Easy
1 Mark
What 2D shapes form the net of a square-based pyramid?
A
1 square and 4 triangles
B
2 squares and 4 rectangles
C
5 triangles
D
1 square and 3 triangles
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
1 square and 4 triangles
Step-by-Step Explanation:
The base is 1 square, surrounded by 4 triangular faces.
13
Multiple Choice
Easy
1 Mark
How many sides does a regular octagon have?
A
8 sides
B
6 sides
C
7 sides
D
10 sides
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
8 sides
Step-by-Step Explanation:
An octagon has 8 sides and 8 angles.
14
Multiple Choice
Easy
1 Mark
Which 3D solid has exactly 1 circular base, 1 curved surface, and 1 apex?
A
Cone
B
Cylinder
C
Sphere
D
Pyramid
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
Cone
Step-by-Step Explanation:
A cone has 1 flat circular base, 1 curved surface, and 1 pointed apex.
15
Multiple Choice
Easy
1 Mark
How many faces, edges, and vertices does a standard triangular prism have?
A
5 faces, 9 edges, 6 vertices
B
6 faces, 12 edges, 8 vertices
C
5 faces, 8 edges, 5 vertices
D
4 faces, 6 edges, 4 vertices
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
5 faces, 9 edges, 6 vertices
Step-by-Step Explanation:
A triangular prism has 2 triangular bases + 3 rectangular sides = 5 faces, 9 edges, and 6 vertices.
16
Short Answer
Hard
2 Marks
A wooden cube has a side length of 4 cm. It is painted red on all outer surfaces and then cut into 1 cm unit cubes. How many 1 cm cubes have exactly 2 red faces?
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
24 cubes
Step-by-Step Explanation:
Cubes with 2 painted faces lie along the 12 edges (excluding corners). Each edge has 4 - 2 = 2 such cubes. 12 edges × 2 = 24 cubes.
17
Short Answer
Hard
2 Marks
Verify Euler's formula (F + V - E = 2) for a pentagonal prism. List its number of faces, vertices, and edges.
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
F = 7, V = 10, E = 15; 7 + 10 - 15 = 2
Step-by-Step Explanation:
Faces = 2 pentagons + 5 rectangles = 7. Vertices = 5 + 5 = 10. Edges = 5 + 5 + 5 = 15. 7 + 10 - 15 = 2.
18
Short Answer
Hard
2 Marks
Why are triangles considered the most rigid and structurally stable geometric shape in engineering bridge trusses?
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
A triangle's shape is completely locked by its side lengths. Unlike rectangles, which can deform into parallelograms under load, a triangle cannot change shape without bending or breaking a member.
Step-by-Step Explanation:
Triangulation provides geometric rigidity because side lengths uniquely fix internal angles.
19
Short Answer
Hard
2 Marks
Explain why a sphere cannot be formed by folding a flat sheet of 2D paper without cutting, stretching, or wrinkling.
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
A sphere has continuous double curvature across its surface, whereas a sheet of paper has zero curvature (flat plane). Flat paper cannot conform to spherical curvature without crumpling.
Step-by-Step Explanation:
Gaussian curvature of a sphere is positive, while paper is flat (zero curvature).
20
Short Answer
Medium
1 Mark
A student draws a net for a cube with 5 connected squares. Why cannot this net fold into a cube?
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
A cube must have 6 faces
Step-by-Step Explanation:
A cube requires exactly 6 square faces; a 5-square net leaves one side missing.
21
Short Answer
Medium
1 Mark
Lucas says: 'A cylinder has 2 edges and 2 vertices.' Explain Lucas's error and state the correct number of vertices on a cylinder.
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
A cylinder has 0 vertices
Step-by-Step Explanation:
The circular edges of a cylinder are smooth loops with no sharp corners (vertices). A cylinder has 0 vertices.
22
Fill in the Blanks
Medium
1 Mark
Euler's formula for polyhedra is Faces + Vertices - Edges = [ ? ].
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
2
Step-by-Step Explanation:
Euler's characteristic formula for convex polyhedra is F + V - E = 2.
23
Fill in the Blanks
Medium
1 Mark
Solid Shape Riddle: 'I have 6 identical flat square faces, 12 equal edges, and 8 corners.' What shape am I?
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
Cube
Step-by-Step Explanation:
A cube has 6 identical square faces, 12 edges, and 8 vertices.
24
Short Answer
Medium
1 Mark
A student glues together two identical wooden cubes face-to-face to form a single cuboid. How many flat faces does the new cuboid have?
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
6 faces
Step-by-Step Explanation:
A cuboid always has 6 flat rectangular faces.
25
Short Answer
Medium
1 Mark
A shipping crate in the shape of a rectangular cuboid has dimensions 5 m long, 3 m wide, and 2 m high. What is the total surface area of all 6 faces?
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
62 sq m
Step-by-Step Explanation:
2 × (5×3 + 5×2 + 3×2) = 2 × (15 + 10 + 6) = 2 × 31 = 62 square meters.
26
Short Answer
Medium
1 Mark
At the Eco-Friendly Solar Park, an architect constructs a geometric installation containing 4 cubes and 3 triangular prisms. How many total flat faces exist across all 7 shapes?
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
39 faces
Step-by-Step Explanation:
4 cubes × 6 faces = 24. 3 triangular prisms × 5 faces = 15. Total = 24 + 15 = 39 flat faces.
27
Multiple Choice
Easy
1 Mark
What 2D shapes form the net of a square-based pyramid?
A
1 square and 4 triangles
B
2 squares and 4 rectangles
C
5 triangles
D
1 square and 3 triangles
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
1 square and 4 triangles
Step-by-Step Explanation:
The base is 1 square, surrounded by 4 triangular faces.
28
Multiple Choice
Easy
1 Mark
How many sides does a regular octagon have?
A
8 sides
B
6 sides
C
7 sides
D
10 sides
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
8 sides
Step-by-Step Explanation:
An octagon has 8 sides and 8 angles.
29
Multiple Choice
Easy
1 Mark
Which 3D solid has exactly 1 circular base, 1 curved surface, and 1 apex?
A
Cone
B
Cylinder
C
Sphere
D
Pyramid
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
Cone
Step-by-Step Explanation:
A cone has 1 flat circular base, 1 curved surface, and 1 pointed apex.
30
Multiple Choice
Easy
1 Mark
How many faces, edges, and vertices does a standard triangular prism have?
A
5 faces, 9 edges, 6 vertices
B
6 faces, 12 edges, 8 vertices
C
5 faces, 8 edges, 5 vertices
D
4 faces, 6 edges, 4 vertices
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
5 faces, 9 edges, 6 vertices
Step-by-Step Explanation:
A triangular prism has 2 triangular bases + 3 rectangular sides = 5 faces, 9 edges, and 6 vertices.
31
Short Answer
Hard
2 Marks
A wooden cube has a side length of 4 cm. It is painted red on all outer surfaces and then cut into 1 cm unit cubes. How many 1 cm cubes have exactly 2 red faces?
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
24 cubes
Step-by-Step Explanation:
Cubes with 2 painted faces lie along the 12 edges (excluding corners). Each edge has 4 - 2 = 2 such cubes. 12 edges × 2 = 24 cubes.
32
Short Answer
Hard
2 Marks
Verify Euler's formula (F + V - E = 2) for a pentagonal prism. List its number of faces, vertices, and edges.
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
F = 7, V = 10, E = 15; 7 + 10 - 15 = 2
Step-by-Step Explanation:
Faces = 2 pentagons + 5 rectangles = 7. Vertices = 5 + 5 = 10. Edges = 5 + 5 + 5 = 15. 7 + 10 - 15 = 2.
33
Short Answer
Hard
2 Marks
Why are triangles considered the most rigid and structurally stable geometric shape in engineering bridge trusses?
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
A triangle's shape is completely locked by its side lengths. Unlike rectangles, which can deform into parallelograms under load, a triangle cannot change shape without bending or breaking a member.
Step-by-Step Explanation:
Triangulation provides geometric rigidity because side lengths uniquely fix internal angles.
34
Short Answer
Hard
2 Marks
Explain why a sphere cannot be formed by folding a flat sheet of 2D paper without cutting, stretching, or wrinkling.
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
A sphere has continuous double curvature across its surface, whereas a sheet of paper has zero curvature (flat plane). Flat paper cannot conform to spherical curvature without crumpling.
Step-by-Step Explanation:
Gaussian curvature of a sphere is positive, while paper is flat (zero curvature).
35
Short Answer
Medium
1 Mark
A student draws a net for a cube with 5 connected squares. Why cannot this net fold into a cube?
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
A cube must have 6 faces
Step-by-Step Explanation:
A cube requires exactly 6 square faces; a 5-square net leaves one side missing.
36
Short Answer
Medium
1 Mark
Lucas says: 'A cylinder has 2 edges and 2 vertices.' Explain Lucas's error and state the correct number of vertices on a cylinder.
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
A cylinder has 0 vertices
Step-by-Step Explanation:
The circular edges of a cylinder are smooth loops with no sharp corners (vertices). A cylinder has 0 vertices.
37
Fill in the Blanks
Medium
1 Mark
Euler's formula for polyhedra is Faces + Vertices - Edges = [ ? ].
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
2
Step-by-Step Explanation:
Euler's characteristic formula for convex polyhedra is F + V - E = 2.
38
Fill in the Blanks
Medium
1 Mark
Solid Shape Riddle: 'I have 6 identical flat square faces, 12 equal edges, and 8 corners.' What shape am I?
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
Cube
Step-by-Step Explanation:
A cube has 6 identical square faces, 12 edges, and 8 vertices.
39
Short Answer
Medium
1 Mark
A student glues together two identical wooden cubes face-to-face to form a single cuboid. How many flat faces does the new cuboid have?
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
6 faces
Step-by-Step Explanation:
A cuboid always has 6 flat rectangular faces.
40
Short Answer
Medium
1 Mark
A shipping crate in the shape of a rectangular cuboid has dimensions 5 m long, 3 m wide, and 2 m high. What is the total surface area of all 6 faces?
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
62 sq m
Step-by-Step Explanation:
2 × (5×3 + 5×2 + 3×2) = 2 × (15 + 10 + 6) = 2 × 31 = 62 square meters.
41
Short Answer
Medium
1 Mark
At the Harbor Marine Terminal, an architect constructs a geometric installation containing 4 cubes and 3 triangular prisms. How many total flat faces exist across all 7 shapes?
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
39 faces
Step-by-Step Explanation:
4 cubes × 6 faces = 24. 3 triangular prisms × 5 faces = 15. Total = 24 + 15 = 39 flat faces.
42
Multiple Choice
Easy
1 Mark
What 2D shapes form the net of a square-based pyramid?
A
1 square and 4 triangles
B
2 squares and 4 rectangles
C
5 triangles
D
1 square and 3 triangles
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
1 square and 4 triangles
Step-by-Step Explanation:
The base is 1 square, surrounded by 4 triangular faces.
43
Multiple Choice
Easy
1 Mark
How many sides does a regular octagon have?
A
8 sides
B
6 sides
C
7 sides
D
10 sides
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
8 sides
Step-by-Step Explanation:
An octagon has 8 sides and 8 angles.
44
Multiple Choice
Easy
1 Mark
Which 3D solid has exactly 1 circular base, 1 curved surface, and 1 apex?
A
Cone
B
Cylinder
C
Sphere
D
Pyramid
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
Cone
Step-by-Step Explanation:
A cone has 1 flat circular base, 1 curved surface, and 1 pointed apex.
45
Multiple Choice
Easy
1 Mark
How many faces, edges, and vertices does a standard triangular prism have?
A
5 faces, 9 edges, 6 vertices
B
6 faces, 12 edges, 8 vertices
C
5 faces, 8 edges, 5 vertices
D
4 faces, 6 edges, 4 vertices
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
5 faces, 9 edges, 6 vertices
Step-by-Step Explanation:
A triangular prism has 2 triangular bases + 3 rectangular sides = 5 faces, 9 edges, and 6 vertices.
46
Short Answer
Hard
2 Marks
A wooden cube has a side length of 4 cm. It is painted red on all outer surfaces and then cut into 1 cm unit cubes. How many 1 cm cubes have exactly 2 red faces?
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
24 cubes
Step-by-Step Explanation:
Cubes with 2 painted faces lie along the 12 edges (excluding corners). Each edge has 4 - 2 = 2 such cubes. 12 edges × 2 = 24 cubes.
47
Short Answer
Hard
2 Marks
Verify Euler's formula (F + V - E = 2) for a pentagonal prism. List its number of faces, vertices, and edges.
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
F = 7, V = 10, E = 15; 7 + 10 - 15 = 2
Step-by-Step Explanation:
Faces = 2 pentagons + 5 rectangles = 7. Vertices = 5 + 5 = 10. Edges = 5 + 5 + 5 = 15. 7 + 10 - 15 = 2.
48
Short Answer
Hard
2 Marks
Why are triangles considered the most rigid and structurally stable geometric shape in engineering bridge trusses?
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
A triangle's shape is completely locked by its side lengths. Unlike rectangles, which can deform into parallelograms under load, a triangle cannot change shape without bending or breaking a member.
Step-by-Step Explanation:
Triangulation provides geometric rigidity because side lengths uniquely fix internal angles.
49
Short Answer
Hard
2 Marks
Explain why a sphere cannot be formed by folding a flat sheet of 2D paper without cutting, stretching, or wrinkling.
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
A sphere has continuous double curvature across its surface, whereas a sheet of paper has zero curvature (flat plane). Flat paper cannot conform to spherical curvature without crumpling.
Step-by-Step Explanation:
Gaussian curvature of a sphere is positive, while paper is flat (zero curvature).
50
Short Answer
Medium
1 Mark
A student draws a net for a cube with 5 connected squares. Why cannot this net fold into a cube?
💡 Click to View Correct Answer & Explanation ▼
Correct Answer:
A cube must have 6 faces
Step-by-Step Explanation:
A cube requires exactly 6 square faces; a 5-square net leaves one side missing.