What is the perimeter of a regular 15- gon with radius of 3?

Answer 1

12

Answer 2

You can work out the length of the edges using the law of cosines. If you break the polygon down into 15 isosceles triangles, they will all have an angle of 360°/15 = 24°, with adjacent side lengths of 3. We can then say:

a2 = b2 + c2 - 2bc cos(A)

In this case, a will be the length of our outside edge:

a2 = 32 + 32 - 2 * 3 * 3 cos(24°)

∴ a2 = 18 - 18cos(24)

∴ a2 ≈ 10.36477786793405443313

∴ a ≈ 3.21943750800260982999

That gives us the length of one side of the polygon, which means that our total perimeter will be 15 times that:

p = 15a

∴ p ≈ 48.29156262003914744985