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# Regular Polygons

With equal angles and equal side lengths, what more could you want from a polygon?

# Regular Polygons: Level 3 Challenges

The diagram above shows a regular hexagon $${ H }_{3 }$$ with area $$H$$ which has six right triangles inscribed in it. Let the area of the shaded region be $$S$$, then what is the ratio $$H:S$$?

Above figure shows a unit square $$ABCD$$.

If the area of the octagon $$EFGHIJKL$$ (in blue) can be expressed as $$\dfrac{1}{a}$$ , find $$a$$.

As shown in the image above, a pentagon, hexagon and decagon are inscribed in three congruent circles, and their endpoints are connected to form a triangle. If the radii of each of the circles is $$1$$ and the area of the triangle formed by the three polygons can be written as $$\frac{\sqrt{a}-b}{c}$$, where $$a$$, $$b$$ and $$c$$ are coprime integers, what is $$a+b+c$$?

We can make "pencilogons" by aligning multiple identical pencils end of tip to start of tip together without leaving any gaps, as shown above, so that the enclosed area forms a regular polygon (the example above left is an 8-pencilogon).

Hazri wants to make an $$n$$-pencilogon using $$n$$ identical pencils with pencil tips of angle $$7^\circ.$$ After he aligned $$n-18$$ pencils, he found out the gap between the two ends is too small to fit in another pencil.

So, in order to complete the pencilogon, he has to sharpen all the $$n$$ pencils so that the angle of all the pencil tips becomes $$(7-m)^\circ$$.

Find the value of $$m+n$$.

Greedy Garisht is celebrating his birthday with 6 of his friends. His mother baked him a birthday cake in the shape of a regular hexagon. Wanting to keep most of the cake, he makes cuts linking the midpoints of every 2 adjacent sides, and distributes these 6 slices to his friends. What proportion of the cake does he have left for himself?

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