Rectangle ABCD is inscribed in a circle. If the radius of the circle is 2 and `bar(CD)=2`, what is the area of the shaded region ?

Options

A. `0.362`
B. `0.471`
C. `0.577`
D. `0.707`

Answer

A

Explanation

As usual, the key here is to add to the figure. And here, as is so often the case with circles, you should add radii. The equilateral triangles formed by `bar(CD)` and the radii, and by `bar(AB)` and the radii, have central angles of `60^(@)`

The shaded region is what's left of the `60^(@)` sector after you subtract the area of an equilateral triangle. To find the area of the shaded region, find the areas of the sector and triangle, then substract the latter from the former. The sector is exactly one - sixth of the circle (because `60^(@)` is one - sixth of `360^(@)`). Thus,
Area of the sector `=(1)/(6)pi r^(2)=(1)/(6)(pi)(2^(2))=(4pi)/(6)`
Use the formula for the area of an equilateral triangle to close in on the answer.
Area of an equilateral triangle `= (s^(2)sqrt(3))/(4)`
`(2^(2)sqrt(3))/(4)=(4sqrt(3))/(4)=sqrt(3)`
The shaded area, then, is `(4pi)/(6)-sqrt(3)~~0.362`.