Area of Square on Hexagon Diagonal

CAT 2017 Slot 2 · QA · Medium · Geometry

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Let ABCDEFABCDEF be a regular hexagon with each side of length 1 cm1\text{ cm}. The area (in sq cm) of a square with ACAC as one side is

  1. A.

    323\sqrt{2}

  2. B.

    3

  3. C.

    4

  4. D.

    3\sqrt{3}

Answer

B

Explanation

In regular hexagon ABCDEFABCDEF, AB=BC=1 cmAB = BC = 1\text{ cm} and ∠ABC=120∘\angle ABC = 120^\circ.

Using Law of Cosines in △ABC\triangle ABC: AC2=AB2+BC2−2(AB)(BC)cos⁡(120∘)AC^2 = AB^2 + BC^2 - 2(AB)(BC)\cos(120^\circ) AC2=12+12−2(1)(1)(−12)=1+1+1=3AC^2 = 1^2 + 1^2 - 2(1)(1)\left(-\frac{1}{2}\right) = 1 + 1 + 1 = 3

Since AC2AC^2 is the area of a square constructed on ACAC, the area is 3 sq cm.

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