A, B, C, D, and E are points on a circle of radius 2 in counterclockwise order. We know AB = BC = CD = DE = 2. Find [ABCDE].
Look at the following :
Let A = (2,0) B =(1 , sqrt 3) D = (-2,0) E = (-1, -sqrt 3)
ABCD forms a trapezoid with bases BC = 2 AD = 4 and height = sqrt (3)
Its area = (1/2)sqrt (3) ( 2 + 4) = 3sqrt 3
And ADE forms a right triangle with base AD = 4 and height sqrt 3
Its area = (1/2) 4 * sqrt 3 = 2sqrt 3
So
[ABCDE ] = 3sqrt 3 + 2 sqrt 3 = 5sqrt 3 units ^2