Question

A square is inscribed in a circle as shown. If the radius of the circle is 9, what is the area of the shaded region, rounded to the nearest hundredth?

Answer 1

Answer:

Area of shaded region is:

A = 92.41

Step-by-step explanation:

(As the diagram is not shown, consider the diagram attached below.)

r = 9

So

Diagonal of the square = 2r = 18

As all angles of square are of 90°

The diagonal is dividing the angles into 2 halves of 45° at point where diagonal is joining the corner.

Taking sine:

$sin\theta=\frac{perpendicular}{hypotenuse}\\perpendicular = s\\hypotenuse = 2r = 18\\\theta = 45^\circ\\Substitute:\\sin45=\frac{s}{18}\\sin45\cdot18=s\\s=12.73$

Area of Circle:

$A=\pi{r}^2\\A=\pi{9}^2\\A=254.34$

Area of Square:

$Area = s\cdot{s}\\Area=12.73\cdot12.73\\Area = 161.93$

Area of Shaded Region:

Area of circle - Area of square

$A = 254.34-161.93\\A=92.41$