please help - area of a dodecagon?
posted by Knights .
The distance between two opposite vertices of the dodecagon is 2. Find the area of the dodecagon.
first, what is the area of a dodecagon and second how to find it with just the distance between two oppositve vertices.
i found that a dodecagon can be divided into 24 triangles: an apothem, half of one of the sides, and half of the distance between two oppositite vertices of the dodecagon. So how do we do this ?
BTW I did not learn trigonometry so this can be figured out through geometry.
Thanks a lot for helping me if you can.
the diagonal is the diameter of the circumscribed circle.
The area of a dodecagon inscribed in a circle of radius R is 3R^2.
So, here R=1, and A = 3
You can find a geometric derivation of the area in Euclid Book IV, but it's heavy going.
Draw a diagram, splitting the dodecagon into 12 congruent isosceles triangles. The vertex angle of one of the congruent isosceles triangles is 30 degrees, so drop an altitude that isn't from the vertex angle's vertex. That forms a 30-60-90 triangle, and the altitude is 1/2. The area of one of those triangles is 1 * 1/2 * 1/2 = 1/4, and the area of the dodecagon is 12 * 1/4 = 3.