NCERT Solutions for Class 7th Maths Chapter 6 Tiling the Entire Plane — In-text Questions

Book page 160 – 162 Updated on2026-09-19

Q1.

Can you think of a shape whose copies can tile the entire plane?

Answer

A square is the easiest one. Lay squares edge to edge in rows and columns and they cover the whole plane, with no gap and no overlap — like floor tiles in a room.

Angles meeting at a corner of the square grid:
four squares meet, each contributing 90°
4 × 90° = 360° = one full turn ✓
No gap, no overlap.

Rectangles work too, and so do parallelograms and any triangle. But the neatest examples are the regular polygons — square, equilateral triangle and regular hexagon.

Why it happens: a tiling has to work at every corner as well as along every edge. At a corner, the angles of the tiles meeting there must add to exactly 360° — less leaves a gap, more forces an overlap. The square passes this test easily, because 90° divides 360° four times over.

Q2.

Are there other regular polygons that can tile the plane? What about equilateral triangles?

Answer

Yes — equilateral triangles and regular hexagons. Those two, together with the square, are the only regular polygons that can tile the plane.

Regular polygon	Each angle	How many fit at a point	Tiles the plane?
Equilateral triangle	60°	360 ÷ 60 = 6	Yes
Square	90°	360 ÷ 90 = 4	Yes
Regular pentagon	108°	360 ÷ 108 = 3.33…	No
Regular hexagon	120°	360 ÷ 120 = 3	Yes
Regular heptagon	≈128.57°	360 ÷ 128.57 = 2.8	No
Regular octagon	135°	360 ÷ 135 = 2.67	No

At the marked point the angles must add to exactly 360°. Squares, triangles and hexagons manage it; pentagons and octagons cannot.

Equilateral triangles: each angle 60°
6 × 60° = 360° ⇒ six triangles fit around every point ✓

Regular hexagons: each angle 120°
3 × 120° = 360° ⇒ three hexagons fit around every point ✓

Regular pentagons: each angle 108°
3 × 108° = 324° (a 36° gap) and 4 × 108° = 432° (an overlap of 72°)
⇒ pentagons cannot tile the plane

Why it happens: for a single regular polygon to tile the plane, its angle must divide 360° exactly a whole number of times. Only 60°, 90° and 120° do that among the regular polygons — every angle from the heptagon onwards is more than 120° but less than 180°, so two are too few and three are too many.

Did you know? Using more than one shape, or non-regular shapes, opens up endless possibilities. The Dutch artist M. C. Escher (1898 – 1972) tiled the plane with birds, fish and lizards. One new tiling was discovered as recently as 2023 — tiling is still an active area of research in geometry.

Q3.

Have you seen tilings in daily life?

Answer

Yes — once you start looking, tilings are everywhere.

Floors and walls: square or rectangular tiles in homes, hexagonal paving blocks on footpaths, brick walls where each brick is offset by half its length.
Rangoli and kolam: repeating units that fill a doorway with no gaps.
Cloth and jaali work: block-printed fabrics and the pierced stone screens of Mughal buildings repeat one motif across a whole surface.
In nature: the front face of a beehive and some wasp nests are tiled with hexagonal cells. The insects store eggs, larvae, pupae and food in them.
Skins and shells: the scales of a fish, the plates on a tortoise shell, the pattern on a pineapple.

Why it happens: a tiled surface wastes no space — that is the practical reason floors and honeycombs are tiled. Three regular hexagons meet at 3 × 120° = 360°, so hexagonal cells pack perfectly with shared walls, which also saves wax. Scientists still wonder exactly how bees and wasps manage to build them so accurately.

Try This: next time you walk past a paved footpath, look at the shape of the blocks and check the angles where three or four of them meet. They will add up to 360°.

Was this helpful?