NCERT Solutions Ganita Prakash (Part 2) Chapter 4 –100Isometric Projections and isometric drawing — In-text Questions

Book page 97 Updated on2026-09-05

Q1.

Construct a model of a cube and use your hands to keep it balanced on one corner vertex. Can you try to understand why all the projected edges have equal length?

Answer

Balance the cube on one corner so that the long diagonal through that corner runs straight up and down. Now project it down onto the floor.

The reason is symmetry. Look at the three edges meeting at the bottom corner. Spin the cube through 120° about the vertical diagonal: the cube lands exactly on itself, and those three edges swap round among themselves. So no one of them is in any way special — they must all be tilted from the vertical by the same angle, and therefore their shadows on the floor must all be the same length.

Take the edge to be 1 unit and the diagonal direction to be along (1, 1, 1)
An edge along (1, 0, 0) makes an angle θ with the diagonal, where cos θ = 1⁄√3
Length of its shadow = sin θ = √(1 − 1⁄3) = √(2⁄3) ≈ 0.816 units

The same number comes out for the other two edges. And every edge of the cube is parallel to one of these three, so all twelve edges project to the same length. That is exactly what the word isometric — ‘equal measure’ — is recording.

Why this is so useful: in a general drawing of a solid you cannot measure anything, because different edges have shrunk by different amounts. In an isometric drawing every edge along the three main directions has shrunk by the same factor. So lengths along those directions can be compared, and counted, straight off the paper. That is why engineers draw on isometric grids.

Q2.

Have you played Tetris? There are five basic shapes in Tetris, corresponding to the different ways of arranging four squares. Imagine these are cubes, not squares. Draw each of these on your isometric paper.

Answer

The five flat shapes are the ways of joining four squares edge to edge (counting a shape and its turned-over copy as the same):

Shape	Arrangement
I	Four in a straight row
O	A 2 × 2 square
L / J	Three in a row with one turning off at the end
S / Z	Two pairs, offset by one — a short zig-zag
T	Three in a row with one attached at the middle

How to draw them. On the isometric grid, use the three directions | for height, one slant for depth and the other slant for length. Then draw one cube at a time, or better, draw the whole outline by counting edges as you go.

Choose which axis the shape will lie along — a row of four looks quite different lying along the depth axis and standing up the height axis.
Draw the outline first, counting one grid unit per cube.
Put in the internal edges that separate the cubes — the lines between the top faces of neighbouring cubes.
Leave out any line that would be hidden inside the solid, or draw it faintly and darken only the visible ones later.

Tip: Shading helps a lot. Colour all the top faces one shade, all the left faces a second and all the right faces a third. The solid then reads instantly, because the three shades correspond to the three axis directions.

Q3.

For example, you can draw a 1 × 1 × 1 cube as follows. How would you draw a 2 × 2 × 2 cube?

Answer

Draw exactly the same hexagonal outline, but make every stroke two grid units long instead of one.

Start at the near bottom corner. Go 2 units along the length direction, then 2 units up, then 2 units along the depth direction, and so on right round the hexagon.
From the topmost corner draw the three internal edges, each 2 units long, meeting at the centre of the hexagon.
If you want the individual small cubes shown, mark the midpoint of every edge and join the midpoints in the three axis directions.

1 × 1 × 1 cube: hexagon outline with sides of 1 unit
2 × 2 × 2 cube: the same hexagon with sides of 2 units — twice as wide, twice as tall
It is built from 2 × 2 × 2 = 8 small cubes, of which 7 are visible from any one direction

Why the drawing simply scales: in an isometric projection every length along the three axes is shrunk by the same factor. Doubling the cube doubles every one of those lengths, so the picture is the same shape at twice the size. If the projection were not isometric, doubling the cube would change the picture in a more complicated way.

Q4.

Why is this correspondence between directions on isometric paper and axes of the solid so effective for communicating the shape of the solid?

Answer

Because two things that could have gone wrong do not.

Parallel stays parallel. The chapter proved that projection takes parallel lines to parallel lines. A cuboid has three families of parallel edges — height, length, depth — so on the paper they become three families of parallel lines: |, and the two slants. Every edge on the paper therefore announces which axis of the solid it belongs to, just by its direction.

Equal stays equal. Because the projection is isometric, a unit step along any of the three axes becomes a step of the same length on the paper. So one grid unit means one cube, whichever direction you count in.

Direction on the paper → axis of the solid (unambiguous)
Number of grid units → number of cubes (unambiguous)

Put together, this means the drawing can be read back: anyone can count edges along the grid and rebuild the solid exactly. That is the whole purpose of an engineering drawing, and an ordinary perspective sketch cannot do it — in perspective, parallel edges meet at a vanishing point and far edges are drawn shorter, so counting is impossible.

The price paid: an isometric drawing looks a little unnatural, because our eyes do see distant things smaller. It trades realism for measurability — and for engineering that is exactly the right trade.

Q5.

Can you try drawing the other tetris shapes on isometric paper?

Answer

Yes — take them one at a time, and for each one first decide the orientation.

Shape	How to lay it out	Edges to count
O (2 × 2 block)	2 along the length, 2 along the height, 1 deep	2, 2, 1
L / J	3 along the depth, then 1 turning off along the length	3 then 1
S / Z	2 along the length, step 1 across, 2 more along the length	2, then 2 offset by 1
T	3 along the length with 1 sticking out at the middle	3 with 1 at the centre

Use the hint the book gives later: before drawing an edge, decide whether it goes from the near end to the far end or the other way, and draw it in the direction of that axis or opposite to it. Drawing the outline first and the internal cube edges afterwards keeps the picture clean.

Try This: Draw the same L-shape three times — once lying along the length axis, once along the depth axis and once standing up the height axis. Three quite different-looking pictures, one solid. This is the best way to convince yourself that direction on the paper really does carry the meaning.

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