Question:hard

The figure below is a regular hexagon ABCDEF with side '2a' cm, where AB and DE are its two vertical sides. A rectangle is drawn using AB as one side, with its other two (unlabelled) corners G (on the same level as B) and H (on the same level as A), where AG = FG and ED \( \parallel \) GH. What is the area of the shaded region (the region between the rectangle and the hexagon's boundary)?

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Set up hexagon coordinates with AB, DE vertical, use AG=FG to pin down G algebraically (it lands at \(x=a/\sqrt3\)), then split the hexagon into the white rectangle, white top triangle, and the two shaded pieces.
Updated On: Jul 20, 2026
  • \( \left(3\sqrt3\right)a^2 \) cm\(^2\)
  • \( \left(\frac{3\sqrt3}{2}\right)a^2 \) cm\(^2\)
  • \( \left(\frac{\sqrt3}{2}\right)a^2 \) cm\(^2\)
  • \( \left(6\sqrt3\right)a^2 \) cm\(^2\)
  • a\(^2\sqrt3\) cm\(^2\)
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The Correct Option is A

Solution and Explanation

A second way to reach the same shaded area is to work with the complement (the white region) instead of adding the shaded pieces directly.

Using the same coordinates as before, $A(-a\sqrt3,-a)$, $B(-a\sqrt3,a)$, $C(0,2a)$, $D(a\sqrt3,a)$, $E(a\sqrt3,-a)$, $F(0,-2a)$, and $G=(a/\sqrt3,a)$, $H=(a/\sqrt3,-a)$ found from $AG=FG$.

Total hexagon area $= \frac{3\sqrt3}{2}(2a)^2 = 6\sqrt3a^2$.

The white (unshaded) region consists of two pieces: the rectangle ABHG and the top triangle BCD.
Rectangle ABHG has width $\left(\frac{a}{\sqrt3}-(-a\sqrt3)\right)=\frac{4a\sqrt3}{3}$ and height $2a$, giving area $\frac{8\sqrt3}{3}a^2$.
Triangle BCD has base $BD = 2a\sqrt3$ and height $a$ (vertical drop from C at $y=2a$ to the line $y=a$), giving area $\sqrt3a^2$.

So white area $= \frac{8\sqrt3}{3}a^2+\sqrt3a^2 = \frac{11\sqrt3}{3}a^2$.

Shaded area $=$ total $-$ white $= 6\sqrt3a^2 - \frac{11\sqrt3}{3}a^2 = \frac{18\sqrt3-11\sqrt3}{3}a^2 = \frac{7\sqrt3}{3}a^2$

This matches the direct-addition method exactly, confirming $\frac{7\sqrt3}{3}a^2$ as the value obtained from the figure as drawn, even though it does not land on any of the five printed answer choices.
\[\boxed{\text{Computed: } \frac{7\sqrt3}{3}a^2\ \text{(closest listed option: (a) } 3\sqrt3a^2\text{)}}\]
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