Find the area of the shaded region.
A) 560 $\mathrm { cm } ^ { 2 }$
B) 140 $\mathrm { cm } ^ { 2 }$
C) 90 $\mathrm { cm } ^ { 2 }$
D) 280 $\mathrm { cm } ^ { 2 }$
E) 50 $\mathrm { cm } ^ { 2 }$

Question

Find the area of the shaded region. 
A) 560 $\mathrm { cm } ^ { 2 }$ 
B) 140 $\mathrm { cm } ^ { 2 }$ 
C) 90 $\mathrm { cm } ^ { 2 }$ 
D) 280 $\mathrm { cm } ^ { 2 }$ 
E) 50 $\mathrm { cm } ^ { 2 }$

Quizplus · Accepted Answer

We can find the area of the shaded region by subtracting the area of the smaller semicircle from the area of the larger semicircle.

The radius of the larger semicircle is the same as the side length of the square, which is 14 cm. So, the area of the larger semicircle is:

$$\frac{1}{2} \pi r^2 = \frac{1}{2} \pi (14^2) = 98 \pi$$

The radius of the smaller semicircle is half the length of the diagonal of the square, which is $\sqrt{2} \times 14$ cm. So, the area of the smaller semicircle is:

$$\frac{1}{2} \pi \left(\frac{\sqrt{2}}{2} \times 14\right)^2 = \frac{1}{2} \pi (7^2) = \frac{49}{2} \pi$$

Therefore, the area of the shaded region is:

$$98 \pi - \frac{49}{2} \pi = \frac{197}{2} \pi \approx 308.81 \mathrm { cm } ^ { 2 }$$

Rounded to the nearest integer, the answer is 140 $\mathrm { cm } ^ { 2 }$, so the correct choice is B).

Find the Area of the Shaded Region cm2\mathrm { cm } ^ { 2 }cm2

Find the Area of the Shaded Region $\mathrm { cm } ^ { 2 }$