*equilateral*. All their angles are the same likewise, which provides them

*equiangular*. For triangles, it transforms out that being

*equilateral*and

*equiangular*constantly go together.

But is that true for other shapes?

## Puzzle 1.

Find a pentagon that is equilateral however NOT equiangular.

You are watching: 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 show us that the civilization of forms is bigger than we imagined!

Another standard array of triangle is the *right* *triangle*. It has one best angle, and also is the basis for trigonomeattempt. (Trigonometry originates from the Greek *tri - *three, *gonna* - angle, and also *metron* - to meacertain.) If we move as much as quadrilateral, it’s simple to discover forms via 4 best angles, namely, rectangles. I deserve to uncover a pentagon via 3 right angles, however not more than that.

## Puzzle 3.

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

A clarification on puzzle 3: we’re only talking around inner appropriate angles right here.

Research question: is tright here some means to predict the maximum number of appropriate angles a polygon can have, once you recognize exactly how many type of sides it has? For instance, deserve to you predict the maximum number of ideal angles 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 point, yet can we prove these are the maximum number of right angles we have the right to fit into each polygon?

We have the right to, if we recognize the formula for the angle amount of polygons: the internal angles of an *n-*gon amount to (n - 2) x 180 levels.

This means that a decagon’s angles sum to 1440 levels. If a decagon had actually 8 appropriate angles, that would account for 720 degrees, leaving 2 angles left to account for the various other 720 levels.

In other words, each of those last angles would have to be 360 degrees. That’s difficult. So a decagon have the right to have actually at a lot of 7 ideal angles. By making the geomeattempt numerical, we can prove what’s true for all shapes, also if there are infinitely many type of. That’s the type of connection that provides mathematics so powerful.