Find the area of the blue shaded region in this 14 by 7 rectangle, with two semicircles of radius 7 drawn.
+128470 See the following image :
If we draw a line segment between the two intersection points of the semi-circles....the length of this segment - AC - is given by
2√ [ 7^2 - 3.5^2] = (2 * 3.5)√ [ 4 - 1] = 7√3
Using the Law of Cosines
(7√3)^2 = 7^2 + 7^2 - 2 (49) cos (ABC)
147 = 98 - 2(49)cos (ABC)
49 = -2(49) cos(ABC)
-1/2 = cos (ABC)
arccos (-1/2) = ABC =
ABC = 120° = (2/3)pi
So (1/2) of the blue shaded area =
Area of sector ABC - area of triangle ABC
(1/2) (7^2)(2/3)pi - (1/2)(7^2) sin (120°)
(1/2)(7^2)(2/3)pi - (1/2)(7^2)*√3/2
(1/2) (7^2) [ (2/3) pi - √3/2 ]
So....twice this area = the blue shaded area =
49 [ (2/3)pi - √3/2 ] units^2
