Geometry
# Regular Polygons

${ H }_{3 }$ 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 $H:S?$

The diagram above shows a regular hexagon

Above figure shows a unit square $ABCD$.

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

*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 aligns $n-18$ pencils, he finds 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 $m+n$.

(Assume the pencils have a rectangular body and have their tips resembling isosceles triangles)