Can two figures (of any kind, bounded by curves or not) have the same area and perimeter, but different shapes?

Question

Can two figures (of any kind, bounded by curves or not) have the same area and perimeter, but different shapes?

By "different shape", I mean that you cannot rotate one, take it out of the plane, and/or enlarge/reduce it so that it fits the other.

score 25 · Answer 1 · answered Sep 05 '16 at 20:49

25

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answered Sep 05 '16 at 20:49

Blue

1

+1 for this elementary, instructive and colourful answer that is only seen once in a Blue moon. (I can guess what answers users Red or Green would have submitted...) – Georges Elencwajg Sep 05 '16 at 21:32
3

@GeorgesElencwajg: So far as colorful answers go, you might say that this isn't just elementary, it's primary. :) – Blue Sep 05 '16 at 22:16
Excellent, dear Blue :-) – Georges Elencwajg Sep 06 '16 at 07:34
1

Hmph. https://xkcd.com/608/ – Lieutenant Zipp Jan 14 '21 at 06:09

score 8 · Answer 2 · answered Sep 05 '16 at 20:34

8

These two figures have the same area ancd the same perimeter:

First figure: the right triangle of sides $3,4,5$.

Second figure: the rectangle of sides $3+ \sqrt{3}, 3- \sqrt{3}$.

answered Sep 05 '16 at 20:34

Crostul

Nice, how you arrived at these values? Interested to know. – Karri Chandrasekhar Nov 18 '21 at 09:13
1

@KarriChandrasekhar First, I considered the triangle of sides 3,4,5. It is easy to compute area and perimeter, since it is a right triangle. The area is $6$ and the perimeter is $12$. After that I solved the quadratic equation $$x^2-6x+6=0$$ the two solutions are two numbers $x_1+x_2=6$ and $x_1x_2=6$, hence the rectagle of sides $x_1, x_2$ has area 6 and perimeter 12. – Crostul Nov 18 '21 at 13:53
Thanks for a detailed explanation. – Karri Chandrasekhar Nov 25 '21 at 12:48