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
[tex] A_{triangle}=\dfrac{1}{2}a^2\sin \alpha [/tex],
where a is length of the side of the octagon and [tex] \alpha [/tex] is the angle between two radii of the octagon.
In regular octagon
[tex] \alpha=\dfrac{360^{\circ}}{8}=45^{\circ} [/tex].
Then the area of regular octagon is
[tex] 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}[/tex] sq. m.
Answer: [tex] A_{octagon}=72\sqrt{2}[/tex] sq. m.
