#1**+2 **

Assuming that this is a regular octagon.......

We can break the octagon into 8 congruent triangles like this:

The sum of the 8 central angles = 360° , and each central angle has the same measure,

so the measure of one central angle = 360° / 8 = 45°

Draw a height of one of the triangles perpendicular to a side of the octagon. I labeled the height "a".

By the HL theorem, the two smaller triangles that are formed are congruent.

So the measure of the angle formed by the height and the hypotenuse = 45° / 2 = 22.5°

tan 22.5° = 1 / a cm

a tan 22.5° = 1 cm

a = 1 / tan 22.5° cm

area of one of the eight triangles = (1/2)(base)(height)

area of one of the eight triangles = (1/2)(2)(1/tan 22.5°) sq cm

area of one of the eight triangles = 1 / tan 22.5° sq cm

area of octagon = 8 * (area of one of the eight triangles)

area of octagon = 8 * (1 / tan 22.5° ) sq cm

area of octagon = 8 / tan 22.5° sq cm

area of octagon ≈ 19.3 sq cm

hectictar
Jun 28, 2018

#1**+2 **

Best Answer

Assuming that this is a regular octagon.......

We can break the octagon into 8 congruent triangles like this:

The sum of the 8 central angles = 360° , and each central angle has the same measure,

so the measure of one central angle = 360° / 8 = 45°

Draw a height of one of the triangles perpendicular to a side of the octagon. I labeled the height "a".

By the HL theorem, the two smaller triangles that are formed are congruent.

So the measure of the angle formed by the height and the hypotenuse = 45° / 2 = 22.5°

tan 22.5° = 1 / a cm

a tan 22.5° = 1 cm

a = 1 / tan 22.5° cm

area of one of the eight triangles = (1/2)(base)(height)

area of one of the eight triangles = (1/2)(2)(1/tan 22.5°) sq cm

area of one of the eight triangles = 1 / tan 22.5° sq cm

area of octagon = 8 * (area of one of the eight triangles)

area of octagon = 8 * (1 / tan 22.5° ) sq cm

area of octagon = 8 / tan 22.5° sq cm

area of octagon ≈ 19.3 sq cm

hectictar
Jun 28, 2018