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NCERT Solutions Ganita Prakash (Part 2) Chapter 4 The octahedron and the dodecahedron — In-text Questions

Book page 83 Updated on2026-09-05

Q1.
Can you visualise its net? This is one of its nets.
Answer

An octahedron has 8 equilateral triangular faces, so its net is made of 8 triangles. The one shown in the book is a strip of 8 triangles arranged as two rows of four, the rows offset by half a triangle.

Octahedron: F = 8, E = 12, V = 6
Check: F + V − E = 8 + 6 − 12 = 2 ✓

How to see it. The octahedron is two square pyramids glued base to base. Unfold the top pyramid outwards to give a ring of 4 triangles, do the same for the bottom pyramid, and slide the two rings together into a zig-zag strip. Every free edge in the strip has exactly one partner to be glued to.

Why the faces meet 4 at a vertex: four equilateral triangles round a point give 4 × 60° = 240°, which is less than 360°, so the flat fan can close into a cone point. That is the vertex where the two pyramids meet. At the apex of each pyramid, the same thing happens.
Q2.
Taking all the triangles in the net to be equilateral, make a cutout of the net and fold it to form an octahedron.
Answer

Use equilateral triangles of side 5 cm.

  1. Draw a strip 8 triangles long, alternately pointing up and down, all with side 5 cm.
  2. Rearrange it into the book’s net: two rows of four, offset so that each triangle in the top row shares a full edge with one below.
  3. Cut it out, adding 1 cm flaps on alternate outer edges.
  4. Fold along every internal line in the same direction and tape the free edges in pairs.
Check it yourself: when it is finished, exactly 4 triangles should meet at each of the 6 vertices, and the solid should have 12 edges. Hold it by two opposite vertices and spin it — a correctly built octahedron spins smoothly, like a top.
Did you know? Like the cube, the octahedron has exactly 11 nets. This is not a coincidence — the cube and the octahedron are ‘dual’ solids: the cube has 6 faces and 8 vertices, the octahedron 8 faces and 6 vertices, and both have 12 edges. The dodecahedron, with 12 pentagonal faces, has 43,380 nets.
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