a dodecon is a 12 sided figure.
each side forma an isosceles triangle with the vertex at the center of the figure which is also the center of a circumscribed circle.
the central angle of each of the triangles is 30 degrees because 360 / 12 = 30.
if you drop a perpendicular from the vertex to the base of each of these triangles, each will form two right triagnles with a 15 degree angle at the vertex point and a 75 degree angle between the radius of the dodecagon and one of its sides.
you can solve for the height of each of these right triangles and you can solve for the length of the base of each of these right triangles.
from that you get the height of each of the 12 triangles formed by the dodecagon and you can solve for the side length of each of the 12 triangles formed by the dodecagon.
from that you can find the area of each of the 12 triangles of the dodecagon and from that you can get the area of the dodecagon.
i did that and came up with 1200 square units as the area of the dodecagon with a radius of 20.
one reference for finding the area of a dodecagon can be found here:
http://www.tutorvista.com/content/math/area-of-a-dodecagon/
an online calculator that will do the calculations for you can be found here:
http://www.cleavebooks.co.uk/scol/calpolyg.htm
this calculator is useful to check you work.
i did the calculations manually as follows:
number of sides of a dodecagon is 12
length of the radius is 12.
central angle of each of the 12 triangle formed is 30 degrees.
drop a perpendicular to the base of each of the isosceles triangles formed andyou get 2 right triangles.
use one of the right triangles to get the length of the altitude and the length of the base of the right triangle.
double the length of the base of the right triangle to get the length of the base of one of the 12 triangles formed.
find the area of one of the triangles.
multiply the area by 12 to get the area of the dodecagon.
i then used the calculator to confirm i got the right answer.
i input 20 for the length of the outside radius and i enter 12 for the number of sides.
the calculator confirmed i did it right.
the calculator calls the side of the dodecagon an edge.
the outside radius is called the circum-radius.
there are several formula that will get you the area of the dodecagon.
the method i used is just one of them.
they are all formed from the same basic information regarding the relationships in the dodecagon.
here's a picture of the output from the calculator.
here's a picture of my manual calculations.
