NCERT Solutions for Class 5th Maths Chapter 7 Three regular pentagons around a point — Tiling and Tessellation
Book page 94 Updated on2026-09-19
Q1.
We have placed 3 pentagons around a point. Can we fit one more into the empty space?
Answer
No, a fourth pentagon will not fit. The empty space left is far too thin for it.
Three corners of a regular pentagon do not quite fill the turn around a point, and what is left is much too small for a fourth pentagon.
Corner of a regular pentagon = 108 Three corners = 108 + 108 + 108 = 324 A full turn round a point = 360 Space left over = 360 − 324 = 36 But one pentagon corner needs 108, and 108 is much more than 36.
So there are only two choices, and both are wrong:
Leave the gap — then the floor has a hole in it, which is not allowed in tiling.
Push a fourth pentagon in — then it climbs on top of the others. That is an overlap, which is also not allowed.
Why it happens: To cover a floor with no gaps, the corners that meet at a point must add up to exactly one full turn. For pentagons the corner size 108 does not divide 360 evenly — 3 corners are too few and 4 are too many. So regular pentagons do not tessellate.
Try it yourself: Cut out five or six pentagons from the sheet at the end of the book and lay them round a dot on your desk. You will see the same thin gap every single time, no matter how you turn them.