Geometry
# Regular Polygons

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)

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