How to show that a languge is not regular?

How to show the regular polygon has the maximum area?

• How can we show that, out of all convex polygons with n ≥ 3 sides and a same perimeter, the regular one has the maximum area? Thank you

• Answer:

  Let the (polar) coordinates of the vertices of the n-gon be (r,t1), (r,t2), ... (r,tn). Divide the n-gon into n triangles with central angles (t2-t1), (t3-t2), (t4-t3), ... (t1-tn). The area of the kth triangle would be (1/2)r²sin(t_(k+1) - t_k). The total area of the n-gon is the sum A = Σ (1/2)r² sin(t_(k+1) - t_k) from k=1 to n (assuming the last term t_(n+1) = t1). Optimize A by taking the first partial derivatives with respect to t_k. We have ∂A/∂t1 = -(1/2)r² cos(t_2 - t_1) + (1/2)r² cos(t_1 - t_n) Set this equal to 0 and we have cos(t_2 - t_1) = cos(t_1 - t_n). Note that both of the arguments in the cosine functions must be positive and are in the interval (0,2pi], and therefore t_2 - t_1 = t_1 - t_n. If you keep taking the other partial derivatives, you would have t_2 - t_1 = t_3 - t_2 t_3 - t_2 = t_4 - t_3 and so on. It follows that the maximum area is attained when all the central angles are equal, which makes the n-gon regular.