SOLUTION: what is the perimeter of a regular dodecagon (a polygon which has 12 sides)whose area is 24 + 12 sqrt(3).

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Question 568980: what is the perimeter of a regular dodecagon (a polygon which has 12 sides)whose area is 24 + 12 sqrt(3).
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
To calculate the area of regular polygon with N sides, the polygon can be sliced pizza-style into N isosceles triangles with one side as the base, and the area of those triangles can be calculated.
The height of those triangles is called the apothem, and can be calculated based on the length of the side, s, and half of the vertex angle, which depends on the number of sides, N.
For a dodecagon, the half of the vertex angle is B=360%5Eo%2F24
Putting it all together, the area, A, of a regular polygon with N sides of length s is
A=s%5E2N%2F%284tan%28pi%2Fn%29%29 or A=s%5E2N%2F%284tan%28180%5Eo%2Fn%29%29 if you prefer your angles in degrees.
With N=12 , A=s%5E2%2A12%2F%284tan%28180%5Eo%2F12%29%29=3s%5E2%2Ftan%2815%5Eo%29 --> Atan%2815%5Eo%29=3s%5E2 --> s%5E2=Atan%2815%5Eo%29%2F3
tan%2815%5Eo%29=2-sqrt%283%29 as can be calculated from tan%2845%5Eo%29 and tan%2830%5Eo%29 using the formula (trigonometric identity) for tangent of a difference.
So s%5E2=Atan%2815%5Eo%29%2F3 with A=24%2B12sqrt%283%29=12%282%2Bsqrt%283%29%29 and tan%2815%5Eo%29=2-sqrt%283%29 gives us

The length of a side of the polygon is s=2
and the perimeter is 12%2A2=highlight%2824%29