# Smallest and largest possible angles of given polygon

What is the smallest and largest possible angle of a triangle? (my guess = 1, 178)

What is the smallest and largest possible angle of a quadrilateral (convex or concave doesn't matter, and also perfect integers (no decimals) only)? (my guess = 1, 357)

If my guesses are right, then:-

The smallest possible angle of an n-sided polygon is always 1.

The largest possible angle of an n-sided polygon is always ((n - 2) * 180) - (n - 1)

Angles must be specified in degrees not radians.

• I suggest we add the constraint that the polygon must be convex. Jul 25 '14 at 14:45
• And as an added challenge: how would the answer differ if we required all of the lengths to be integers also? Jul 25 '14 at 14:47
• I suggest you to improve the math formatting, explain that the angles are given in degrees, the $n$-sided polygons you consider are only the convex one and so on. Jul 25 '14 at 14:48
• @ ozo, this answer is in the question. Jul 25 '14 at 14:53

The set of values which are valid angle measurements does not have a minimum or a maximum if we are talking about non-degraded convex polygons. It does have an infimum and supremum - in case of a triangle the infimum is 0 and the supremum is 180.

I think, your guess is wrong. You guess smallest possible angle is 1 and largest possible angle is 178 for triangle. Triangle with this angles have angles 1, 1, 178 as you know the sum of all angles of a triangle is 180. We can take an angle which is less than 1. Let this angle be 0.1. Then the sum of other two angles must be 179.9, these angles can be 179 and 0.9 . So I 've proved your assumption wrong by counterexample. Actually, We can't decide smallest or largest angles for a triangle or polygon.

• Lets not consider decimals. Sorry for not informing. Jul 25 '14 at 15:21

Well that is not the case . The largest possible value of an interior angle could be is 180 . After that I think , it is denoted as an exterior angle . Just think about it a little bit . Also if n is supposed to be 5 then largest angle according to the above formula is 536. Not possible .

• After an interior angle goes above 180, it becomes concave. I don't mind concave quadrilaterals. Oh yes, I forgot! Angles above 359 don't make sense. Jul 25 '14 at 15:22

Angles do not usually have to be an integral number of degrees, so if you are requiring that you should say so. In that case, for the triangle you are fine. For the quadrilateral, once the angle gets larger than $180$ you have a concave quadrilateral, which is not always allowed, but if you accept them, you are fine. For more sides, angles greater than $359$ do not make sense.

• 'integral number of degrees' What does that mean? Yes, concave quads are allowed. Oh yes, I forgot! Angles above 359 don't make sense. Jul 25 '14 at 15:23
• That you don't allow $0.1$ degree angles, for example. Usually they are allowed. Some of the other answers focus on that. Jul 25 '14 at 15:31
• Okay, yeah so only integral angles allowed. Jul 25 '14 at 15:44