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CAT 2021 Slot 1 — QA Question 9

Basics of Mensuration/PrismEasy

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

Answer & solution

  • A

    4√6

  • B

    √6

  • C

    6√6

  • 2√6

Solution

Easy

A regular hexagon is six equilateral triangles of its own side. Write both areas with the 34 (side)2\tfrac{\sqrt3}{4}\,(\text{side})^2 formula, set them equal, and solve for the hexagon's side.

hexagon = 6 equilateral triangles of side a triangle, side 12
1

Area of the equilateral triangle. Side 1212.

A△=34×122=363\begin{aligned} &A_\triangle=\frac{\sqrt3}{4}\times 12^{2}=36\sqrt3 \end{aligned}
2

Area of the regular hexagon. It is six equilateral triangles of side aa.

Ahex=6×34 a2\begin{aligned} &A_{\text{hex}}=6\times\frac{\sqrt3}{4}\,a^{2} \end{aligned}
3

Equate the areas and solve. Set Ahex=A△A_{\text{hex}}=A_\triangle.

6×34 a2=34×122(from steps 1,2)⇒ 6a2=144⇒ a2=24⇒ a=24=26\begin{aligned} &6\times\frac{\sqrt3}{4}\,a^{2}=\frac{\sqrt3}{4}\times 12^{2} \quad\text{(from steps 1,2)}\\ &\Rightarrow\ 6a^{2}=144\\ &\Rightarrow\ a^{2}=24\\ &\Rightarrow\ a=\sqrt{24}=2\sqrt6 \end{aligned}
Side of hexagon=26 cm(option (d))\text{Side of hexagon}=2\sqrt6\ \text{cm} \quad\text{(option (d))}
CAT 2021 Slot 1 QA Q9: If the area of a regular hexagon is equal to the area of an equilateral triangle of side 12 cm, then the lengt — Solution | TheCATExam