If all the sides of an n sided polygon are equal. Is it always a regular polygon?

Question

For an $n$-sided polygon, if all the sides are equal, is it a regular polygon?
If yes then why is it defined to have equal angles?
If not so, how to prove that all angles are equal?

edit: I meant it for $n>4$

edit 2: I tried to play with some figures

A regular polygon is one which is both equilateral and equiangular. For $n>3$, these terms do not coincide. — lc2r43, Jul 26 '21 at 06:27
@J.W.Tanner,What could be an example for number of sides greater than 4? — Dheeraj Gujrathi, Jul 26 '21 at 06:30
A parallelogram which is not a square but have all equal sides. — mathcounterexamples.net, Jul 26 '21 at 06:33
@mathcounterexamples.net,could you please give me an example for number of sides greater than 4? — Dheeraj Gujrathi, Jul 26 '21 at 06:36
@bernard,could you please explain me what is complete definition of a regular polygon,and why is it so? — Dheeraj Gujrathi, Jul 26 '21 at 06:44
All sides have to be equal, all angles equal and convex. Alternative definition, for a polygon with $n$ sides: it has $n$ symmetry axes, or it is invariant by a rotation of order $n$. — Bernard, Jul 26 '21 at 06:51
@DheerajGujrathi Play with making pentagons on a table, out of same-length toothpicks. Do you think they must all be regular pentagons? — dxiv, Jul 26 '21 at 06:53
@dxiv,i got a fair idea after trying to play with sticks,but wouldn't be there any case i have not attached them edge to edge?I have attached an image of what I tried to do,so finally there exists a convex polygon,having all sides equal,but with unequal angles.. — Dheeraj Gujrathi, Jul 26 '21 at 07:15
@DheerajGujrathi See also Does specifying the lengths of the sides of a polygon completely fix its shape (area and angles)? — dxiv, Jul 26 '21 at 07:20
Since your "edit 2" (Jul 26, 2021 at 7:12) , there was no need left of an answer: one of the two equilateral pentagons of your picture is not equiangular. — Anne Bauval, Sep 14 '23 at 08:44

score 7 · Accepted Answer · answered Jul 26 '21 at 06:37

7

The equiangular part of the definition is not a redundancy, if that is what you mean. As a counterexample consider the polygon that appears on the Swiss flag. It is an equilateral dodecagon, 12 sides, all equal in length. It is not equiangular. Eight of the interior angles are 90°, and the other four are 270°. It has only four rotation symmetries and four reflections. A regular dodecagon has 12 of each.

answered Jul 26 '21 at 06:37

Pope

656

ohhh yes,,thanks a lot,that is why I love stack community – Dheeraj Gujrathi Jul 26 '21 at 06:38
is the convexity of regular polygon a result of the property or is a part of the definition? – Dheeraj Gujrathi Jul 26 '21 at 06:51
3

@DheerajGujrathi Ask yourself if a non-convex polygon can have all angles equal. – dxiv Jul 26 '21 at 06:58
@dxiv,ohh yess,they can't as one of them would have angle greater than 180,for eg,182,so all of them should have angle 182,which isn't possible to get a closed figure,thanks – Dheeraj Gujrathi Jul 26 '21 at 08:18

ThePretzelMan · Answer 2 · 2021-07-27T14:15:01.767

2

For polygons with more then 3 sides, it is not always a regular polygon. Assume a rhombus, which is a type of parallelogram that has the added restriction of being equilateral. It could have angles that don't all measure 90° (eg two 45° angles and two 135° angles).

That's the case for $n=4$, but what about the rest? For $n>4$ a more general method of generating irregular equilateral polygons would be to have one angle be 90°, and the rest will have measures according to this formula: $$\frac{90(2n-3)}{n-1}$$ Where n is the number of sides of the polygon.

edited Jul 27 '21 at 14:15

answered Jul 26 '21 at 06:48

ThePretzelMan

352

,I tried using that formula,but It didn't seem to work,or I didn't knew where to use it and how,could you give me an example? – Dheeraj Gujrathi Jul 26 '21 at 07:48
I had accidentally typed a 5 instead of a 3 in the numerator. The formula should be accurate now. The way to use the formula would to have one of the angles be 90°, and the formula dictates the angle for all others. – ThePretzelMan Jul 27 '21 at 14:16
,ok,I would try it for sure,thanks – Dheeraj Gujrathi Jul 27 '21 at 16:18

If all the sides of an n sided polygon are equal. Is it always a regular polygon?

2 Answers2