Let $ABCDEF$ be a regular hexagon with each side of length $1$ cm. The area (in sq cm) of a square with $AC$ as one side is

A regular hexagon is a 6-sided polygon where all sides are equal and all interior angles are equal. Each interior angle of a regular hexagon measures 120°.

We need to find the length of diagonal AC in regular hexagon ABCDEF, then calculate the area of a square with AC as one side.

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We focus on triangle ABC:

AB = 1 cm (given side of hexagon)

BC = 1 cm (given side of hexagon)

∠ABC = 120° (interior angle of regular hexagon)

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The cosine rule states: $c^2 = a^2 + b^2 - 2ab \cos(C)$

In triangle ABC:

$AC^2 = AB^2 + BC^2 - 2(AB)(BC)\cos(∠ABC)$

$AC^2 = 1^2 + 1^2 - 2(1)(1)\cos(120°)$

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Since $120° = 180° - 60°$, we have:

$\cos(120°) = -\cos(60°) = -\frac{1}{2}$

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$AC^2 = 1^2 + 1^2 - 2(1)(1)\left(-\frac{1}{2}\right)$

$AC^2 = 1 + 1 - 2\left(-\frac{1}{2}\right)$

$AC^2 = 2 + 1 = 3$

Therefore: $AC = \sqrt{3}$ cm

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A square with side length $\sqrt{3}$ cm has area:

Area = $(\text{side})^2 = (\sqrt{3})^2 = 3$ cm²

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Answer: 3 cm²

This approach is efficient because we only need to analyze one triangle rather than the entire hexagon, the cosine rule directly gives us the diagonal length, and regular hexagon properties ensure all similar diagonals are equal.