+0

​how

0
299
1
+214

howhow

Jun 28, 2018

#1
+8146
+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

Jun 28, 2018
edited by hectictar  Jun 28, 2018

#1
+8146
+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
edited by hectictar  Jun 28, 2018