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)=(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=(360^(\circ))/(8)=45^(\circ)`.

Then the area of regular octagon is

`A_(octagon)=8A_(triangle)=8\cdot (1)/(2)a^2\sin \alpha=4\cdot (6)^2\cdot \sin 45^(\circ)=4\cdot 36\cdot (√(2))/(2)=72√(2)` sq. m.

Answer:
`A_(octagon)=72√(2)` sq. m.