A Regular Hexagon is Inscribed in a Circle, What is the Ratio of the Area of the Hexagon GMAT Problem Solving

Question: A regular hexagon is inscribed in a circle, what is ratio of the area of the hexagon to the area of the circle?

1. √3 : \(\pi\)
2. 2√3 : 3\(\pi\)
3. 3√3 : \(\pi\)
4. 3√3 : 2\(\pi\)
5. √3 : 5\(\pi\)

‘A regular hexagon is inscribed in a circle, what is the ratio of the area of the hexagon to the area of the circle?’ - is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “Princeton Review GMAT Premium Prep”. 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.

Solution and Explanation:

Approach Solution 1:

There is only one approach solution to this problem.
It is given in the question that a regular hexagon is inscribed in a circle. It has to find out the ratio of the area of the hexagon to the area of the circle.
To solve this question let us assume that the side of the hexagon is 4 units.
Side of hexagon = 4
Now we know that a hexagon is inscribed in the circle so the diameter of the circle will be the longest diagonal in the hexagon.
It should be noted that a regular hexagon is composed of six equilateral triangles of equal sides.
Hexagon = 6 equilateral triangles of same side.
Also, we can say that diameter of the circle = 2 * side of the hexagon.
Diameter of a circle = 2 * 4 = 8
The area of the hexagon will be 6 times the area of an equilateral triangle.
Area of the hexagon = 6 *(\(\sqrt{3}\)/4 \(a^2\))
= 6 *((\(\sqrt{3}\)/4) \((4)^2\)) = 6 *(\(\sqrt{3}\)/4 *16) = 24\(\sqrt{3}\)
Area of circle =\(\pi\) \(r^2 \)=\(\pi\) \((4)^2\) = 16\(\pi\)
The ratio of the area of the hexagon and the circle =
Area of hexagon / area of circle = 24\(\sqrt{3}\) / 16\(\pi\) = 3\(\sqrt{3}\) / 2\(\pi\)
Therefore the correct answer will be option D.

Correct Answer: D

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