The distance between two opposite vertices of the dodecagon is 2. Find the area of the dodecagon.
No use of trigonometry. I know this formula which states that the area of the dodecagon is three times the circumscribed circle's radius but you aren't allowed to use that.
My dad gave me this problem and I'm stumped.
Aug 15th 2013, 11:59 AM
It looks straight forward to me. I assume, of course that you mean a regular dodecahedron since otherwise this is impossible. If the distance between opposite vertices is 2, then the distance from center to each vertex is 1. So you have 12 isosceles triangle with side lengths 1 and vertex angle 360/12= 30 degrees. Dividing each isosceles triangle in half gives 24 right triangles with hypotenuse of length 1 and angle 30 degrees. Can you find the lengths of the legs, so the area of each right triangle, and the multiply by 24?
Aug 15th 2013, 12:13 PM
Um, the right triangles would form a 15, 75, 90 degree right triangle, not 30.
Aug 15th 2013, 07:03 PM
I don't know if the attachment shows a proof acceptable to your dad, but at least it doesn't explicitly use trig.