NCERT Solutions Ganita Prakash (Part 2) Chapter 4 Nets of other solids — In-text Questions

Book page 81 Updated on2026-09-05

Q1.

What is a net of a regular tetrahedron? Which of the following are nets of a regular tetrahedron?

Answer

A net of a regular tetrahedron is four equilateral triangles joined edge to edge so that they fold up into the solid. The first figure (the big triangle) and the third figure (the parallelogram) are nets. The second and the fourth are not.

Figure	What it is	Net?
1st	A large equilateral triangle cut by its midlines into 4 small triangles	Yes
2nd	A strip of 3 triangles with the 4th hanging below the end triangle	No
3rd	A strip of 4 triangles forming a parallelogram	Yes
4th	Five triangles — a strip of 4 with one more hanging below	No

Why the fourth is out at once: a tetrahedron has only 4 faces, and that figure has 5 triangles. One triangle would have nowhere to go.

Why the second fails. Look at the strip of three, up–down–up. All three meet at one point, and each contributes 60°, so the angles round that point add to 180°. Folding closes that point into a vertex of the tetrahedron, and doing so brings the two outer edges of the strip together — those are the two halves of the long base line. The fourth triangle in that figure is stuck onto one of exactly those two edges, so folding would make it collide with the other. It has been attached to the wrong edge.

3 triangles round a point: 60° + 60° + 60° = 180°
180° < 360°, so the flat fan folds into a cone point — a vertex
The 4th triangle must fill the triangular hole left over, so it must be joined to one of the three free edges of the fan, not to the pair that get glued

The only two nets of a regular tetrahedron. In the left one the fourth triangle sits on the middle triangle of the strip; in the right one the strip simply carries on.

Q2.

Are there any other possible nets?

Answer

No — a regular tetrahedron has exactly 2 nets, the big triangle and the parallelogram.

Here is the argument. Whatever the net, it is a chain or fan of 4 triangles. Start from the strip of three (up–down–up), which is forced: any three triangles joined in a line look like that. Its free edges are the two slanting outer edges and the top edge of the middle triangle — but folding glues the two halves of the base together, so the three edges available for the fourth triangle are:

the top edge of the middle triangle — this gives the big triangle net;
the outer edge of the left triangle, or
the outer edge of the right triangle — either of these gives the parallelogram, and the two are mirror images, so they count as the same net.

Why so few: a cube has 11 nets and a tetrahedron only 2 because the tetrahedron has just 4 faces and every face touches every other face. There is almost no freedom left in how you may unfold it.

Q3.

Draw a net with appropriate measurements that can be folded into a regular tetrahedron. Verify if it works by making an actual cutout.

Answer

Take the edge to be 6 cm. Then a net is a single equilateral triangle of side 12 cm, with its three midpoints joined.

Big triangle: each side 12 cm
Mark the midpoint of each side (at 6 cm)
Join the midpoints — this makes 4 identical equilateral triangles of side 6 cm
Fold up along the three midlines; the three corners meet at the apex

How to draw it accurately. Draw a 12 cm segment. With the compass set at 12 cm, cut arcs from both ends; their meeting point is the third vertex. Join up, mark the midpoints with a ruler and join them.

Check it yourself: before folding, measure — all six little edges of the inner triangle and the corner triangles must read 6 cm, and every angle 60°. After folding, the three corner vertices should meet exactly at a point with no gap and no overlap. If there is a gap, your midpoints were not accurate.

Tip: Add 1 cm flaps on alternate outer edges so you can glue the model. The flaps are for building; they are not part of the net.

Q4.

Draw a net with appropriate measurements that can be folded into a square pyramid. Verify if it works by making an actual cutout.

Answer

Take a square base of side 6 cm and slant edges of 8 cm. The net is that square with an isosceles triangle built outwards on each of its four sides.

Square base: 6 cm × 6 cm
Each triangle: base 6 cm, the other two sides 8 cm each
Slant height (height of a triangular face) = √(8² − 3²) = √55 ≈ 7.4 cm

Net of a square pyramid: the base square with four congruent isosceles triangles folded up around it.

Why any slant edge longer than 3√2 cm works. The apex must sit above the centre of the square, and the centre is at a distance of half the diagonal, 6√2⁄2 = 3√2 ≈ 4.24 cm, from each corner. So the slant edge has to be more than 4.24 cm. With 8 cm the pyramid stands up tall:

Height of the pyramid = √(8² − (3√2)²) = √(64 − 18) = √46 ≈ 6.8 cm

Check it yourself: fold the four triangles up. Their apexes should meet at a single point, and the four slant edges should close with no gap. If they do not meet, the four triangles were not congruent.

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