Dividing the pentagon

Question

It is easy to divide an equilateral triangle into three equal, though not equilateral, triangles.

It is even simpler to divide a square into four equal squares.

The difficult part is, whether you can divide a regular pentagon into five equal pentagons?

Note:

Equal means, equal in area.

can we have 1 extra pentagon in the end or the question is about exactly 5? — manshu, Feb 24, 2016 at 19:21

Community · Accepted Answer · 2020-06-17 08:22:35Z

24

Easy, the only definition of a pentagon is that is must have 5 sides :)

You could replicate it for orders 7,9, 11 and so on. (Just figured it won't work for even numbers)

CommunityBot

1

answered Feb 24, 2016 at 19:32

phenxd

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1

$\begingroup$ +1 but how can i believe that their areas are equal $\endgroup$
– manshu
Feb 24, 2016 at 19:33
1

$\begingroup$ Haha, I was just in the process of drawing exactly this, then saw this answer appear. Nice one. $\endgroup$
– jhabbott
Feb 24, 2016 at 19:34
2

$\begingroup$ @manshu if you add 5 points somewhere on the straight lines from the centre to the 5 vertices (doesn't matter where, somewhere near the middle), then rotate all 5 of those elbow points around the centre, then all 5 shapes will be identical (and therefore have the same area). $\endgroup$
– jhabbott
Feb 24, 2016 at 19:36
1

$\begingroup$ @manshu because all of the areas are symmetrical - if all the red lines are drawn in the exact same way and are simply $72\unicode{xb0}$ rotations of each other $\endgroup$
– Paul Evans
Feb 24, 2016 at 19:42
10

$\begingroup$ "Just figured it won't work for even numbers" - then you make the cuts in the middle of the original polygon's edges, like the example with the square, instead of at the vertices. $\endgroup$
– user2357112
Feb 24, 2016 at 23:14

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