A rhombus is inscribed in a circle with radius r and center O such that one of the vertices lies on O and rest lie on the circumference of the circle.
Find the area of the rhombus in terms of r.
Let the long diagonal of the rhombus = L
By the intersecting chord theorem we have that
(3/2)r * (1/2)r = [ ( 1/2) L ]^2
(3/4)r^2 = (1/4)L^2 multiply both sides by 4
(3)r^2 = L^2 take the square root
(√3) r = L
So.....the area of the rhombus = (1/2) product of the diagonals =
(1/2) (r) (√3) r =
Circle's radius > r = 2
Rhombus side > s = r
--II-- long diagonal > d1 =? d1 = sqrt[s2- (r/2)2]*2 d1 = 3.464
--II-- short diagonal > d2 = r
Rhombus area > A = ? A = 1/2(d1*r) or A = s * sqrt(3) or A = (d1*d2)/2
A = 3.464 u2