Brainsteem Mathematics Challenges: Rectangle in a Polygon

rycharde (63) in mathematics • 6 years ago (edited)

Question

The image below shows a regular 14-sided polygon. The shaded rectangle is created from two opposite sides of the shape.

What is the ratio of the area of the rectangle to the area of the whole polygon?

Write the answer as p:q where p and q are coprime.

As always, this is adapted from a Mathematics Competition.

I'm not sure I wish to give a hint...
;-)

erm... the hint is that, being a competition question, calculators are not allowed.

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Enjoy!

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6 years ago by rycharde (63)

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markgritter (59) · 6 years ago

The polygon has area 7/2 * s^2 * cot( pi /14), where s is the side length.

The rectangle has height s * cot( pi /14), so area s^2 * cot(pi/14)

We can get both these from the formula for the length of the apothem which is s / (2 * tan( pi / N ) ) for an N-sided polygon. This is half the height of the rectangle, and it is the height of each of the 14 triangles with base s that make up the polygon.

So, the ratio is 7/2 : 1, that is, 7 : 2.

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rycharde (63) · 6 years ago

That's great, but, just for educational purposes of our readers, there is a geometric method without the need for the trigonometry formulas.

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markgritter (59) · 6 years ago

But we invented trigonometry (and algebra) so we didn't have to be clever about geometry any more! :)

I actually started out with a triangle dissection but I couldn't figure out the trick showing that you could pack exactly four triangles into the rectangle.

Now that I say that, it's probably something like using "all triangles with the same height and base have the same area" but I tend not to think in terms of non-rigid transformations.

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Brainsteem Mathematics Challenges: Rectangle in a Polygon