Question

Consider the region formed by the intersections of three semi circles whose diameters, of lengths 3, 4 and 5, from a triangle. What is the area of shaded region bounded by the arcs AB, BC and AC?

• A. 6
• B. 12
• C. 3π
• D. 6π

Answer 1

The correct option is A

6

A Radius of semicircles ACB, AC and BC are 2.5, 2 and 1.5 respectively.
Area of Semi circles (ACB), (AC) and (BC) are $\frac{6.25\pi }{2}$, 2$\pi$ and $\frac{2.25\pi }{2}$ respectively. And area of
triangle ABC =$\frac{1}{2}$ $\times$ 4 $\times$ 3 = 6
Sum of the area of region A and B = (Area of semicircle (ACB) - Area of triangle (ABC)} = $(\frac{6.25\pi }{2}-6)$
Area of Shaded region = (Sum of the area of semicircles AC and BC - Sum of the area of region A and B)
Area of Shaded region = $(2\pi +\frac{2.25\pi }{2})$ - $(\frac{6.25\pi }{2}-6)$ = $\left(\frac{6.25\pi }{2}\right)$ - $(\frac{6.25\pi }{2}-6)$ = 6

Alternate Solution:
As the options are far apart we can easily arrive at the answer by approximation. The shaded
region is approximately half of the areas of semicircles AC and BC = $\frac{1}{4}(4\pi +2.25\pi )$ = 6(approximately)