Equal Area of Hexagon and Triangle

CAT 2021 Slot 2 · QA · Medium · Geometry

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If the area of a regular hexagon is equal to the area of an equilateral triangle of side 12 cm12\text{ cm}, then the length, in cm, of each side of the hexagon is

  1. A.

    464\sqrt{6}

  2. B.

    666\sqrt{6}

  3. C.

    262\sqrt{6}

  4. D.

    6\sqrt{6}

Answer

C

Explanation

Area of equilateral triangle of side s=12 cms = 12\text{ cm}: AreaΔ=34×122=363\text{Area}_{\Delta} = \frac{\sqrt{3}}{4} \times 12^2 = 36\sqrt{3}

Area of regular hexagon of side aa: Areahex=6×34a2=332a2\text{Area}_{\text{hex}} = 6 \times \frac{\sqrt{3}}{4} a^2 = \frac{3\sqrt{3}}{2} a^2

Given Areahex=AreaΔ\text{Area}_{\text{hex}} = \text{Area}_{\Delta}: 332a2=363  ⟹  a2=24  ⟹  a=24=26 cm\frac{3\sqrt{3}}{2} a^2 = 36\sqrt{3} \implies a^2 = 24 \implies a = \sqrt{24} = 2\sqrt{6}\text{ cm}

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