If you spend too much time v triangles, you deserve to miss just how odd polygons can behave when they have actually a few more sides. Because that example, it is provided triangles have actually all congruent political parties - it is the meaning of equilateral. All your angles are the very same also, which makes them equiangular. Because that triangles, it transforms out the being equilateral and equiangular always walk together.



But is the true for various other shapes?

Puzzle 1.

Find a pentagon that is equilateral however NOT equiangular.

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Puzzle 2.

Find a pentagon that is equiangular but NOT equilateral.

It’s funny to look because that these type of counterexamples. They display us the the human being of shapes is bigger than we imagined!

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

Puzzle 3.

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

A clear up on puzzle 3: we’re only talking around interior best angles here. 

Research question: is over there some means to guess the maximum variety of right angles a polygon have the right to have, as soon as you know how plenty of sides it has? because that example, have the right to you predict the maximum number of right angle a 30-gon have the right to have?


Puzzle 1.

Here is one instance of a pentagon the is equilateral yet not equiangular.

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Finding examples is one thing, yet can us prove these space the maximum number of right angles we have the right to fit into each polygon?

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

This way that a decagon’s angles amount to 1440 degrees. If a decagon had 8 best angles, that would account because that 720 degrees, leaving two angles left come account because that the various other 720 degrees.

In other words, each of those last angles would should be 360 degrees. That’s impossible. Therefore a decagon can have at most 7 right angles. By do the geometry numerical, we have the right to prove what’s true for every shapes, even if there room infinitely many. That’s the type of connection that makes mathematics so powerful.