If you spend too much time with triangles, you have the right to miss exactly how odd polygons deserve to behave once they have a couple of more sides. For example, it is intended triangles have all congruent sides - it is the definition of equilateral. All their angles room the very same also, which makes them equiangular. Because that triangles, it transforms out that being equilateral and equiangular always walk together.


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But is that true for various other shapes?

Puzzle 1.

Find a pentagon the is equilateral but NOT equiangular.

You are watching: An equiangular hexagon that is not equilateral

Puzzle 2.

Find a pentagon that is equiangular but NOT equilateral.

It’s fun to look for these kinds of counterexamples. They display us that the civilization of shapes is bigger 보다 we imagined!

Another an essential variety that triangle is the right triangle. It has actually one right angle, and also is the basis because that trigonometry. (Trigonometry originates from the Greek tri - three, gonna - angle, and metron - to measure.) If we relocate up come quadrilateral, it’s straightforward to find shapes with 4 right angles, namely, rectangles. I can find a pentagon with three right angles, yet not more than that. 

Puzzle 3.

What’s the maximum variety of right angle a hexagon deserve to have? What about a heptagon? an octagon? A nonagon? A decagon?

A clarify on puzzle 3: we’re only talking about interior ideal angles here. 

Research question: is over there some way to guess the maximum number of right angles a polygon have the right to have, when you understand how numerous sides the has? because that example, can you guess the maximum variety of right angle a 30-gon have the right to have?

Solutions

Puzzle 1.

Here is one instance of a pentagon that is equilateral but not equiangular.

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Finding examples is one thing, however can we prove these are the maximum variety of right angles we can fit into each polygon?

We can, if we recognize the formula because that the angle amount of polygons: the interior angles of one n-gon amount to (n - 2) x 180 degrees.

This method that a decagon’s angles amount to 1440 degrees. If a decagon had 8 ideal angles, that would account for 720 degrees, leaving 2 angles left to account for the various other 720 degrees.

In other words, each of those last angle would need to be 360 degrees. It is impossible. For this reason a decagon have the right to have at many 7 right angles. By make the geometry numerical, we can prove what’s true for every shapes, even if there room infinitely many. It is the sort of connection that makes mathematics so powerful.