If you invest too lot time with triangles, you have the right to miss just how odd polygons can behave as soon as they have a few more sides. For example, it is provided triangles have all congruent political parties - it is the definition of equilateral. All their angles space the very same also, which makes them equiangular. For triangles, it turns out the being equilateral and equiangular always walk together.


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But is that true for other shapes?

Puzzle 1.

Find a pentagon that is equilateral however NOT equiangular.

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Puzzle 2.

Find a pentagon the is equiangular but NOT equilateral.

It’s fun to look for these kinds of counterexamples. They present us the the human being of shapes is bigger than we imagined!

Another fundamental variety that triangle is the right triangle. It has one ideal angle, and is the basis because that trigonometry. (Trigonometry originates from the Greek tri - three, gonna - angle, and metron - to measure.) If we relocate up to quadrilateral, it’s basic to discover shapes with four right angles, namely, rectangles. Ns can uncover a pentagon v three best angles, but not much more than that. 

Puzzle 3.

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

A clear up on puzzle 3: we’re just talking about interior right angles here. 

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

Solutions

Puzzle 1.

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

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Finding examples is one thing, however can we prove these are the maximum number of right angle we deserve to fit right into each polygon?

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

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

In various other words, every of those last angle would must be 360 degrees. That’s impossible. For this reason a decagon deserve to have at most 7 appropriate angles. By do the geometry numerical, we can prove what’s true for every shapes, also if there are infinitely many. It is the type of connection that makes mathematics so powerful.