Are you looking for detailed, step-by-step verified solutions for DAV Class 7 Maths Chapter 14 Brain Teasers, HOTS & Value Based Questions (Visualising Solids)? You’ve arrived at the ultimate resource! NCERTBooksPDF provides complete, curriculum-aligned answers for 2D vs 3D shapes, Euler’s formula for polyhedra (F + V – E = 2), nets of solids, isometric vs oblique sketches, top/front/side views, and real-world architectural modeling applications.
DAV Class 7 Maths Chapter 14: all worksheets All Class 7 Maths chapters →
Summary of Polyhedra & Euler’s Formula (Quick Reference)
| Solid (Polyhedron) | Faces (F) | Vertices (V) | Edges (E) | Euler’s Formula (F + V – E) |
|---|---|---|---|---|
| Cube / Cuboid | 6 | 8 | 12 | 6 + 8 – 12 = 2 |
| Triangular Prism | 5 | 6 | 9 | 5 + 6 – 9 = 2 |
| Triangular Pyramid (Tetrahedron) | 4 | 4 | 6 | 4 + 4 – 6 = 2 |
| Square Pyramid | 5 | 5 | 8 | 5 + 5 – 8 = 2 |
| Pentagonal Prism | 7 | 10 | 15 | 7 + 10 – 15 = 2 |
Chapter 14 Brain Teasers Solutions
Q.1 (A). Tick the correct option:
(a) Euler’s formula for any polyhedron with F faces, V vertices, and E edges is:
(i) F + V + E = 2
(ii) F + V – E = 2
(iii) F – V + E = 2
(iv) F + E – V = 2
Answer: (ii) F + V – E = 2
Explanation: For any simple convex polyhedron, the number of faces plus vertices minus edges always equals 2.
(b) A polyhedron having 20 faces and 12 vertices has how many edges?
(i) 28
(ii) 30
(iii) 32
(iv) 24
Answer: (ii) 30
Explanation: Using Euler’s formula: F + V – E = 2 → 20 + 12 – E = 2 → 32 – E = 2 → E = 30.
(c) Which of the following solid shapes has NO vertices and NO flat edges?
(i) Cone
(ii) Cylinder
(iii) Sphere
(iv) Prism
Answer: (iii) Sphere
Explanation: A sphere has only 1 continuous curved surface, 0 vertices, and 0 edges.
(d) The net of a right circular cylinder consists of:
(i) Two rectangles and one circle
(ii) One rectangle and two congruent circles
(iii) One triangle and two circles
(iv) Two circles only
Answer: (ii) One rectangle and two congruent circles
Explanation: Unrolling the curved lateral surface produces a rectangle (of length 2πr and breadth h), while the top and bottom lids are two congruent circular bases.
(e) The number of faces in a triangular pyramid (tetrahedron) is:
(i) 3
(ii) 4
(iii) 5
(iv) 6
Answer: (ii) 4
Explanation: A triangular pyramid has 1 triangular base and 3 triangular lateral faces, making a total of 4 triangular faces.
Q.1 (B). Answer the following questions:
(a) Can a polyhedron have 10 faces, 20 edges, and 15 vertices? Give reasons.
Solution:
Apply Euler’s formula:
F + V – E = 10 + 15 – 20 = 25 – 20 = 5.
Since 5 ≠ 2, this combination violates Euler’s formula.
Therefore, no such polyhedron can exist.
(b) Name the 3D solid that has 1 curved surface and 2 flat circular faces.
Solution:
A Cylinder (right circular cylinder).
(c) How many edges and vertices does a triangular prism possess?
Solution:
A triangular prism has:
• 6 vertices
• 9 edges (3 basal + 3 top + 3 vertical connecting edges)
• 5 faces (2 triangular bases + 3 rectangular lateral faces).
(d) What is the main difference between an Oblique sketch and an Isometric sketch?
Solution:
• Oblique Sketch: Drawn on square grid paper; the front face is drawn to exact scale, but receding depth lines are drawn at 45° and usually foreshortened.
• Isometric Sketch: Drawn on isometric dot paper (dots arranged in equilateral triangles); all three dimensions (length, breadth, and height) are drawn to exact proportional measurements.
(e) What cross-section do you get when you give a (i) vertical cut, (ii) horizontal cut to a cylindrical pipe?
Solution:
(i) A vertical cut produces a Rectangle.
(ii) A horizontal cut produces a Circle (or concentric rings for hollow pipe).
HOTS (Higher Order Thinking Skills) Questions
HOTS 1: Three identical cubes, each of edge length 3 cm, are placed side by side to form a single cuboid. Find:
(a) The dimensions of the resulting cuboid.
(b) The number of faces, edges, and vertices of the new solid.
(c) The total surface area of the resulting cuboid.
Solution:
(a) Dimensions of the cuboid:
Length l = 3 + 3 + 3 = 9 cm.
Breadth b = 3 cm.
Height h = 3 cm.
(b) Elements of the solid:
Since the resulting solid is a cuboid:
• Faces (F) = 6
• Vertices (V) = 8
• Edges (E) = 12.
(Note: The touching faces are now internal and no longer count as boundary faces).
(c) Total Surface Area:
TSA = 2(lb + bh + hl)
= 2[(9 × 3) + (3 × 3) + (3 × 9)]
= 2(27 + 9 + 27) = 2(63) = 126 cm2.
HOTS 2: A regular polyhedron has 30 edges and 12 vertices. How many faces does it possess, and what is the shape of each face?
Solution:
By Euler’s formula:
F + V – E = 2
F + 12 – 30 = 2
F – 18 = 2 → F = 20 faces.
This regular polyhedron is known as an Icosahedron. Each of its 20 faces is an equilateral triangle.
Value Based Questions (VBQs)
Question: In a sustainable architecture project, students built a scaled 3D model of an eco-friendly prefabricated shelter. The structure consists of a cuboidal living unit surmounted by a triangular prism roof.
(a) Count the total number of exterior faces, vertices, and edges of this combined model, and verify whether it satisfies Euler’s formula.
Solution:
• Exterior Faces (F): 1 bottom face + 4 vertical walls + 2 sloping roof faces + 2 triangular gable ends = 1 + 4 + 2 + 2 = 9 faces.
• Vertices (V): 4 bottom corners + 4 wall-top corners + 2 roof ridge endpoints = 4 + 4 + 2 = 10 vertices.
• Edges (E): 4 bottom edges + 4 vertical wall posts + 4 wall-top edges + 4 sloping rafters + 1 top ridge beam = 4 + 4 + 4 + 4 + 1 = 17 edges.
Checking Euler’s Formula:
F + V – E = 9 + 10 – 17 = 19 – 17 = 2.
Euler’s formula is perfectly verified!
(b) Why do architects and structural engineers construct 3D models and unfolding nets before building real structures?
Solution:
1. Error prevention & structural safety: Identifies design flaws before expensive manufacturing.
2. Resource efficiency: Accurate nets minimize raw material wastage during panel fabrication.
3. Client visualization: Helps non-specialists understand living spaces clearly.
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