Question

Find the area of an octagon with a radius of 6 meters

Answer 1

A regular octagon has all its eight sides congruent. The line segments joining each of the vertices of a regular octagon to its center are called the radii of the octagon. These 8 radii divide a regular octagon into 8 congruent isosceles triangles. Area of each isosceles triangle is

$A_{triangle}=\dfrac{1}{2}a^2\sin \alpha$,

where a is length of the side of the octagon and $\alpha$ is the angle between two radii of the octagon.

In regular octagon

$\alpha=\dfrac{360^{\circ}}{8}=45^{\circ}$.

Then the area of regular octagon is

$A_{octagon}=8A_{triangle}=8\cdot \dfrac{1}{2}a^2\sin \alpha=4\cdot (6)^2\cdot \sin 45^{\circ}=4\cdot 36\cdot \dfrac{\sqrt{2}}{2}=72\sqrt{2}$ sq. m.

Answer: $A_{octagon}=72\sqrt{2}$ sq. m.