Vardaan
Class 7 Maths • Chapter 15

Visualising Solid Shapes

Vardaan Learning Institute • Detailed Notes with Practice Questions

🧊 1. 2D vs 3D Shapes

2D (Plane/Flat) shapes have only length and width. Example: square, triangle, circle.
3D (Solid) shapes have length, width, AND height/depth. Example: cube, sphere, cylinder.

Key Terms:
• Face — flat surface of a solid
• Edge — line where two faces meet
• Vertex (plural: vertices) — corner point where edges meet
CubeF=6, E=12, V=8
All faces = squares
CuboidF=6, E=12, V=8
Rectangular faces
CylinderF=3 (2 flat+1 curved)
E=2, V=0
ConeF=2 (1 flat+1 curved)
E=1, V=1
SphereF=1 (curved)
E=0, V=0
Triangular PrismF=5, E=9, V=6
Square PyramidF=5, E=8, V=5

📊 2. Key Table — Properties of 3D Shapes

Shape Faces (F) Edges (E) Vertices (V) F+V−E
Cube 6 12 8 2 ✓
Cuboid 6 12 8 2 ✓
Triangular Prism 5 9 6 2 ✓
Square Pyramid 5 8 5 2 ✓
Triangular Pyramid (Tetrahedron) 4 6 4 2 ✓
Cylinder 3 2 0 1 (curved F)
Cone 2 1 1 2
Sphere 1 0 0 1
Euler's Formula: F + V − E = 2 (for any polyhedron)
Euler's Formula Examples:
• Cube: 6 + 8 − 12 = 2 ✓
• Triangular Pyramid: 4 + 4 − 6 = 2 ✓
• If a solid has F=10, V=6, find E: 10 + 6 − E = 2 → E = 14

📦 3. Nets of 3D Shapes

A net is the 2D flat layout that folds up to form a 3D solid. Multiple nets are possible for some solids.

Net of a Cube: There are 11 different nets of a cube! The classic one is a cross shape with 6 squares.

Net of a Cuboid: Has 6 rectangles in a specific arrangement.
Net of a Cylinder: 2 circles + 1 rectangle (rectangle length = circumference of circle).
Net of a Cone: 1 circle (base) + 1 sector of a bigger circle (curved surface).
Net of a Square Pyramid: 1 square + 4 triangles.

Key Test: A flat shape is a valid net if all faces have correct size and the shape folds into the 3D solid without any overlaps or gaps.

👁️ 4. Views of Solid Shapes

When we look at a 3D solid from different directions, we see different 2D shapes called views:

• Front View: What you see from the front
• Side View: What you see from the left or right side
• Top View: What you see from directly above (also called "Plan")

Example — Cube looked at from:
• Front = Square | Side = Square | Top = Square

Example — Cylinder:
• Front = Rectangle | Side = Rectangle | Top = Circle

Example — Cone:
• Front = Triangle | Side = Triangle | Top = Circle
✏️ Practice Questions — Visualising Solid Shapes (22 Questions)

Section A — Basics Easy

Q1. Name the solid with F=6, E=12, V=8 where all faces are squares
Q2. A square pyramid has how many triangular faces?
Q3. Verify Euler's formula for a triangular prism (F=5, E=9, V=6)
Q4. How many faces does a triangular pyramid (tetrahedron) have?
Q5. Name the 3D shape that looks like a circle from the top and a rectangle from the side.
Q6. How many flat faces does a cone have?
Q7. What is the top view of a cube?
Q8. Name a solid that has no edges and no vertices.

Section B — Euler's Formula & Properties Medium

Q9. A polyhedron has F=7, E=15. Find V using Euler's formula.
Q10. A polyhedron has F=20, V=12. Find E.
Q11. Can a polyhedron have F=3, E=3, V=2? Verify using Euler's formula.
Q12. A pentagonal prism: find F, E, V. Verify Euler's formula.
Q13. A hexagonal pyramid: find F, E, V. (base is hexagon)
Q14. What is the net of a cylinder made of? Describe fully.

Section C — Views and Nets Medium

  1. Q15. Describe the front, side, and top views of a cone placed on its base.
  2. Q16. An ice-cream cone is placed upside down. What are its three views now?
  3. Q17. A net has a rectangle flanked by two circles of matching radius. Which solid does it form?
  4. Q18. Which of these nets will form a cube? (Students to test by drawing and folding mentally — describe a cross-shaped, an L-shaped, and a straight-line arrangement.)

Section D — Challenge Hard

  1. Q19. A solid has a square base and 4 triangular faces meeting at a point. What solid is it? Find F, E, V and verify Euler's formula.
  2. Q20. A sports stadium has a cylindrical body topped with a hemispherical dome. Describe what the front view and top view of this structure look like.
  3. Q21. Is the following true or false: "Two different solids can have the same front view." Give an example to justify your answer.
  4. Q22. If a cube has edge length 4cm: (a) How many unit cubes (1cm each) fit inside it? (b) What is the surface area? (Use: Surface area of cube = 6a²)
✅ Key Answers: Q1:Cube | Q2:4 triangles | Q3:5+6-9=2 ✓ | Q4:4 | Q5:Cylinder | Q6:1 | Q7:Square | Q8:Sphere | Q9:V=10 | Q10:E=30 | Q11:No (3+2-3=2 ✓ mathematically, but not constructible as convex polyhedron) | Q12:F=7,E=15,V=10 | Q13:F=7,E=12,V=7 | Q19:Square Pyramid, F=5,E=8,V=5 | Q22:(a)64 unit cubes, (b)96cm²

📝 Quick Revision

  1. 3D solids have Face (F), Edge (E), Vertex (V). 2D shapes are flat.
  2. Euler's Formula for polyhedra: F + V − E = 2
  3. Cube: F=6, E=12, V=8. Square Pyramid: F=5, E=8, V=5
  4. Cylinder has 2 flat + 1 curved face, 2 edges, 0 vertices
  5. Net = 2D "unfolding" that folds back into 3D solid. Cube has 11 different nets.
  6. Views: Front view / Side view / Top view give 2D cross-sections of 3D solids
  7. Sphere: no edges, no vertices. Cone: 1 vertex, 1 curved face, 1 flat face