Let us call the point where a line from Q drops on to the line OR at an angle of 90 degrees as P. We have to find the perimeter of the figure PQO.



We now have a triangle PQO, which by the given measures is a 30-60-90 triangle.

Now the sides of a 30-60-90 triangle are in the ratio \(x:\sqrt{3}x:2x\) (short side:long side:hypotenuse)

Now we know that the hypotenuse OQ is nothing but the radius and is thus equal to 4 and is equal to 2x in the above ratio. This means that the value of x is 2. From this the we know that the value of the long side OP is \(\sqrt{3}*2 = 2*\sqrt{3}\). And the length of the short side PQ is x, which is equal to 2.


OQ= 4


OP = \(2\sqrt{3}\)



PQ = 2



Now we the know value of one side of the shaded region - PQ = 2.

Now to calculate the value of PR.

From the figure, PR = OR - OP

OR = 4 (the radius)
OP = \(2\sqrt{3}\) (the long side of triangle PQO)

PR = OR - OP
PR = \(4 - 2\sqrt{3}\)



Now we know the value of one more side - PR - of the shaded region PQR.

Now we need to know the value of the third side of the shaded region - the curve QR.


This can be easily found out since we are given the diameter of the circle PS = 8. This means the radius is 4 and the circumference is \(2*\pi*4 = 8*\pi.\)

Now we want to know the value of the central angle QOR. We know this to be 30 degs since it is a part of 30-60-90 triangle.

Now the length of the Arc QR can be found out from the ratio

\(\dfrac{30}{360} = \dfrac{x}{8*\pi}\)

\(\dfrac{1}{12} = \dfrac{x}{8*\pi}\)

\(x = \dfrac{8*\pi}{12} \)

\(x = \dfrac{2*\pi}{3}\)

\(QR = \dfrac{2*\pi}{3}\)


Now we can find the perimeter of the shaded region

PQR = PQ + PR + QR = \(2 + 4 - 2\sqrt{3} + \dfrac{2*\pi}{3} = \dfrac{2*\pi}{3} - 2\sqrt{3} + 6\)

The answer is Choice D - \(\dfrac{2*\pi}{3} - 2\sqrt{3} + 6\)