Question

A regular octagon has the same area as a regular hexagon that has side length 16. How long is each side of the octagon?

Answer 1

Split up the hexagon into six congruent triangles. Each triangle has area
\[A_\Delta=\frac{1}{2}bh\]
where the base is given as 16. Finding the height requires some trigonometry:
\[\tan60^\circ=\frac{h}{8}~~\Rightarrow~~h=4\sqrt3\]
So the area of any one triangle is
\[A_\Delta=\frac{1}{2}(16)(4\sqrt3)=32\sqrt3\]
There are six triangles in the hexagon, so the total area of the hexagon is
\[A=6A_\Delta=192\sqrt3\]