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].

Guest Dec 14, 2021

#1**+1 **

If A, B, C, D, E all lie on the same circle, and the distances between each consecutive point is the same, you know that these five points constitute a regular pentagon inscribed within a circle. Using this, try to find [ABCDE]

Hint: Try drawing triangles from each side length of the pentagon to the middle of the triangle. You should get five congruent isosceles triangles that you can use to find the area.

KnockOut Dec 14, 2021

#2**+1 **

Circle centre O with r = 2, angle subtended between A and B, B and C, C and D, D and E, E and A with centre O = 360^{0} / 5 = 72^{0}

Area of triangle ABO = (1/2)(2*2*sin 72^{0}) = 1.902 sq units

[ABCDE] = There are 5 triangles of equal area = (1.902 sq units)5 = 9.51 sq units.

Guest Dec 14, 2021