Question

an equilateral triangle and a regular hexagon have equal perimeters. if the area of the triangle is 22, then the area of the hexagon is

Answer 1

The area of the hexagon is 33 units square

Given,

An equilateral triangle and a regular hexagon have equal perimeters.

Consider the side of the the equilateral triangle as 2x and side of the hexagon is 2

Then,

3×2x=6×x

6x=6x

Area of the equilateral triangle =$\frac{\sqrt{3} }{4}a^{2}$

Substitute the values

$\frac{\sqrt{3} }{4}(2x)^{2} =22\\\frac{\sqrt{3} }{4}4x^{2}=22\\ \sqrt{3}x^{2} =22\\ x^{2} =\frac{22}{\sqrt{3} }$

Equilateral triangle and a regular hexagon have equal perimeters

We know a regular hexagon consist of 6 equilateral triangles

Area of the regular hexagon = 6 × Area of the equilateral triangles

$=6\frac{\sqrt{3} }{4} x^{2} \\=6\frac{\sqrt{3} }{4}.\frac{22}{\sqrt{3} }$

=33 units square

Hence, the area of the hexagon is 33 units square

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