Question

Find the area of each regular polygon.

12. 18gon with perimeter 72mm.

Answer 1

The central angle of the regular 18-gon is 360/18 = 20.
Because the perimeter is 72 and the 18-gon is regular,
then 72/18 yields 4 for each of the 18 sides.
To find the apothem, note that it is the distance from the center of the polygon to one of the sides.
The apothem is also the altitude of one of the 18 isosceles triangles.
Therefore, it bisects the vertex angle of the triangle as well as the base.
tan (10 degrees) = 2/a
(tan (10) ) * a = 2
a = 2 / [tan(10)]
a = 11.343 approx
-----
A = (1/2) (a) (p)
A = (1/2) (11.343) (72)
A = 408. 332

Answer 2

Thank you so much for this, but I just want to know, from where did that 10 came?

Answer 3

@kiara0497 --> If you followed the 20 degree central angle, [The central angle of the regular 18-gon is 360/18 = 20] then when the apothem was drawn to the base of the isosceles triangle, the apothem bisected that 20 degree angle. That is because the altitude of an isosceles triangle bisects the vertex angle of the triangle.

Answer 4

Here's a link to a resource for study about regular polygons and their areas.

http://www.mathsisfun.com/geometry/regular-polygons.html

Answer 5

Oh okay, thanks a lot!