A regular hexagon is inscribed in a circle. If the length of the shortest diagonal of the hexagon is 12 cm, fi

Question

A regular hexagon is inscribed in a circle. If the length of the shortest diagonal of the hexagon is 12 cm, find one-sixth of the area (in cm²) of the region lying between the circle and the hexagon.

Examoogle · Accepted Answer

The correct answer is Option B: $4(2\pi - 3\sqrt{3})$. Remember that the side of a regular hexagon is the radius of its circumcircle. The shortest diagonal $d = s\sqrt{3} = 12$ implies $s^2 = 48$. One-sixth of the difference is simply $\frac{\pi s^2 - (3\sqrt{3}/2)s^2}{6}$.

Step-by-Step Solution & Concepts:
Shortcut Trick: Remember that the side of a regular hexagon is the radius of its circumcircle. The shortest diagonal $d = s\sqrt{3} = 12$ implies $s^2 = 48$. One-sixth of the difference is simply $\frac{\pi s^2 - (3\sqrt{3}/2)s^2}{6}$.

Formula Used: $$Area_{diff} = Area_{circle} - Area_{hexagon}$$, $$d = s\sqrt{3}$$, $$Area_{hexagon} = \frac{3\sqrt{3}}{2}s^2$$, $$Area_{circle} = \pi s^2$$

Given: A regular hexagon is inscribed in a circle, and the shortest diagonal (d) of the hexagon is 12 cm.

Steps: Find side length of the hexagon: For a regular hexagon of side $s$, the shortest diagonal connects non-adjacent vertices, given by the formula $d = s\sqrt{3}$. Given $d = 12$ cm, we solve for $s$: $s\sqrt{3} = 12 \implies s = \frac{12}{\sqrt{3}} = 4\sqrt{3}$. Thus, $s^2 = (4\sqrt{3})^2 = 16 	imes 3 = 48$.. Calculate the difference in areas: The radius of the circumcircle $R$ is equal to the side $s$ of the hexagon. The area of the circle is $A_c = \pi R^2 = \pi s^2 = 48\pi$. The area of the regular hexagon is $A_h = \frac{3\sqrt{3}}{2}s^2 = \frac{3\sqrt{3}}{2}(48) = 72\sqrt{3}$.. Calculate one-sixth of the area: We need $\frac{1}{6}$ of the region between them: $\frac{1}{6}(48\pi - 72\sqrt{3}) = \frac{48\pi}{6} - \frac{72\sqrt{3}}{6} = 8\pi - 12\sqrt{3}$. Factoring out 4, we get $4(2\pi - 3\sqrt{3})$.

Answer: B is correct because the calculated value $\frac{1}{6}(48\pi - 72\sqrt{3}) = 4(2\pi - 3\sqrt{3})$ matches Option B.

Exam Tip: This type of problem frequently appears in geometry sections of competitive exams; always relate the hexagon's side to the circle's radius first.

Examoogle · Answer

Option A: $6(3\pi - 2\sqrt{3})$

Examoogle · Answer

Option C: $4(3\pi - 2\sqrt{3})$

Examoogle · Answer

Option D: $6(2\pi - 3\sqrt{3})$

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