The Figure Below Shows a Square Inscribed in a Circle with the Radius of \(\sqrt{6}\). What is the Area of the Shaded Region?

Question: The figure below shows a square inscribed in a circle with the radius of \(\sqrt{6}\) . What is the area of the shaded region?

1. \(\pi\)
2. \(6\pi-\sqrt{12}\)
3. \(6\pi-2\sqrt{12}\)
4. \(3({\frac{\pi}{2}}-1)\)
5. \(3{\frac{\pi}{4}-\frac{3}{2}}\)

“The figure below shows a square inscribed in a circle with the radius of \(\sqrt{6}\) . What is the area of the shaded region?” – is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

Answer:

Have a look at the diagram given in the question, we observed that the area of the shaded region is ¼ th of the area of the circle minus area of the square.

Firstly we will find the area of the circle

Area of the circle = \(\pi{r^2}=6\pi\)

Since,

Diagonal of the square = Diameter of the circle

Then value of diagonal = \(2\sqrt6\)

So, Area of the square = \(\frac{(diagonal)^2}{2}\) = 12;

Therefore,

Area of the shaded region =\(\frac{1}{4}(6\pi-12)\) = \(3({\frac{\pi}{2}}-1)\)

Correct option: D

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