The diameter for the smaller semicircle is 2r, find the expression for the shaded area? Now let the diameter of the larger semicircle be 5 calculate the area of the shaded area?

Jun 8, 2018

color(blue)("Area of shaded region of smaller semicircle"=((8r^2-75)pi)/8

color(blue)("Area of shaded region of larger semicircle"=25/8 "units"^2

Explanation:

$\text{Area of } \Delta O A C = \frac{1}{2} \left(\frac{5}{2}\right) \left(\frac{5}{2}\right) = \frac{25}{8}$

$\text{Area of Quadrant } O A E C = {\left(5\right)}^{2} \left(\frac{\pi}{2}\right) = \frac{25 \pi}{2}$

$\text{Area of segment} A E C = \frac{25 \pi}{2} - \frac{25}{8} = \frac{75 \pi}{8}$

$\text{Area of Semicircle} A B C = {r}^{2} \pi$

Area of shaded region of smaller semicircle is:

$\text{Area } = {r}^{2} \pi - \frac{75 \pi}{8} = \frac{\left(8 {r}^{2} - 75\right) \pi}{8}$

Area of shaded region of larger semicircle is area of triangle OAC:

${\text{Area" = 25/8 "units}}^{2}$