In the figure above, AB and CD are diameters of the circle with centre as O, AEB is arc of circle with centre as D and AFB is an arc of the circle with centre as C. If AB = 20cm, what is the area of the shaded region?

(A) 50
(B) 100
(C) 150
(D) 200
(E) 250

Good Question... Kudos to you.. This question tests many Concepts.

For the arc AEB with Centre D, AD = ED = R (Radius of Bigger Circle in for which AEB is an arc).
r = Radius of smaller circle with Centre O.
Now, as AB is the Diameter of Circle with centre O, it will subtend 90 Degree at point C and D.
From Triangle BOD, we have -
\(R^2 = r^2 + r^2\) -----> \(R = \sqrt{2}r.\)

Now, lets find the Area of AEBOA = Area of SECTOR with Centre D - Area of Triangle ADB
Area of AEBFA = \(\frac{90}{360}*pi*R^2 - \frac{1}{2}*2r*r = pi*\frac{r^2}{2} - r^2\)

Now, multiply this area with 2 to get the total area of arc AEBFA = \(pi*r^2 - 2r^2\)
So, area of shaded region = Area of circle with Centre O - Area of unshaded region AEBFA = \(pi*r^2 - (pi*r^2 - 2r^2)\)
Area of shaded region = \(2r^2\) = 200.
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