If you spend too much time with triangles, you can miss how odd polygons can behave once they have a few more sides. For example, equilateral triangles have all congruent sides - that’s the definition of equilateral. All their angles are the same also, which makes them equiangular. For triangles, it turns out that being equilateral and equiangular always go together.


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But is that true for other shapes?

Puzzle 1.

Find a pentagon that is equilateral but NOT equiangular.

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Puzzle 2.

Find a pentagon that is equiangular but NOT equilateral.

It’s fun to look for these kinds of counterexamples. They show us that the world of shapes is bigger than we imagined!

Another fundamental variety of triangle is the right triangle. It has one right angle, and is the basis for trigonometry. (Trigonometry comes from the Greek tri - three, gonna - angle, and metron - to measure.) If we move up to quadrilateral, it’s easy to find shapes with four right angles, namely, rectangles. I can find a pentagon with three right angles, but not more than that. 

Puzzle 3.

What’s the maximum number of right angles a hexagon can have? What about a heptagon? An octagon? A nonagon? A decagon?

A clarification on puzzle 3: we’re only talking about interior right angles here. 

Research question: is there some way to predict the maximum number of right angles a polygon can have, once you know how many sides it has? For example, can you predict the maximum number of right angles a 30-gon can have?

Solutions

Puzzle 1.

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

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

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

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

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