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Regular Polygons

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

Level 3

         

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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