NCERT Solutions for Class 5th Maths Chapter 7 A pattern with more than one shape, and the regular octagon — Tiling and Tessellation
Book page 95 Updated on2026-09-19
Q1.
Here is a tessellating pattern with more than one shape. What shapes have been used in this pattern? _____________, _____________.The tiling pattern given in the book, page 95.
Answer
Regular hexagons and equilateral triangles.
Look at the picture in the book. The big orange shapes have six equal sides — those are regular hexagons. The blue shapes between them have three equal sides — those are equilateral triangles.
Orange hexagons with blue triangles filling every gap. No holes, no overlaps.
At every meeting point: 2 hexagon corners + 2 triangle corners = 120 + 120 + 60 + 60 = 360 — exactly one full turn ✓
Why two shapes are used: Hexagons alone tessellate, and triangles alone tessellate. Here they are mixed, and the mix still works because the corners that meet still add up to 360. A tiling may use more than one shape as long as that rule is kept.
Look for it: This pattern is common on rangoli designs, dupatta prints and old tile floors. Once you know the two shapes, you can spot it everywhere.
Q2.
Continue the pattern given below and colour it appropriately.The part of the pattern printed in the book, page 95.
Answer
Keep adding a purple hexagon with six yellow triangles around it, again and again. The little piece drawn in the book is the start of a hexagon-and-triangle tiling.
Steps
Look at what is drawn. Each purple hexagon has yellow triangles sitting on its sides.
Pick a free side of a hexagon. Put a yellow triangle on it, with its side exactly along the hexagon's side.
Where two triangles point at each other, draw a new purple hexagon in the space between them.
Repeat until the dotted box is full.
Colour all the hexagons in one colour and all the triangles in another. Same shape, same colour.
The finished pattern: purple hexagons, yellow triangles, no gaps anywhere.
Why it never leaves a gap: At every corner two hexagons and two triangles meet. 120 + 120 + 60 + 60 = 360, one full turn. Because the rule is the same at every corner, the pattern can go on for ever.
Colouring tip: Colour by shape, not by position. All hexagons one colour, all triangles another. Then the pattern shows up clearly from across the room.
Q3.
A regular octagon means a shape with eight equal sides. Do regular octagons fit together without any gaps or overlaps? Try drawing the same and check.
Answer
No. Regular octagons do not tessellate. Wherever two of them meet, a square hole is left behind.
Two octagon corners do not fill the turn, and a third octagon is far too big for what is left.
Corner of a regular octagon = 135 Two corners = 135 + 135 = 270, gap left = 360 − 270 = 90 Three corners = 135 × 3 = 405, which is more than 360 → they would overlap So neither 2 nor 3 octagons work.
Why it happens: 360 divided by 135 is not a whole number. Two octagons leave a gap of 90, and a third cannot squeeze into 90 because it needs 135. So a floor of octagons alone always has holes.
But there is a trick: The leftover gap is exactly 90 — the corner of a square. So if you use octagons and small squares together, the floor is covered perfectly. Many old verandah floors are tiled exactly like this. Try drawing it.