In a rectangle ABCD of length 14 and breadth 7, an arc with centre A and radius 14 is drawn. Find the area of shaded region.
Let E be the point on the circle that also lies on DC
Then DE = sqrt ( 14^2 - 7^2) = sqrt [ 7^2 ( 2^2 - 1) ] = 7sqrt (3)
So the area of triangle DEA = (1/2) (DE)(DA) = (1/2) (7sqrt (3)) ( 7) = (49/2)sqrt (3) units^2 (1)
And the sine of angle AED = sine of angle EAB = 7/14 = 1/2
So
arcsin ( 7/14) = arcsin (1/2) = 30° = angle EAB
So.....the area of sector AEB = pi (r^2) (30/360) =
pi ( 14)^2 (1/12) = (196/12) pi = (49/3) pi units^2 (2)
And the area of the rectangle = 14 * 7 = 98 units^2 (3)
So....the shaded area = (3) - (2) - (1) =
98 - (49/2)sqrt (3) - (49/3) pi ≈ 4.252 units^2