Question

An equilateral triangle and a regular hexagon have equal perimeters. if the area of the triangle is 2, what is the area of the hexagon?

Answer 1

`A_h=3 " units"^2`

Explanation:

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Formula for the area of an equilateral triangle with side length `a` is `A_t=sqrt3/4*a^2`
Let `2x` be the side length of the equilateral triangle,
given that area of the equilateral `A_t=2 " units"^2`
`=> A_t=sqrt3/4*(2x)^2=sqrt3/4*4x^2=2`
`=> x^2=2/sqrt3 " units"^2`

A regular hexagon can be divided into 6 congruent equilateral triangles, as shown in the figure.
given that the equilateral triangle and the regular hexagon have equal perimeter,
`=>` side length of the hexagon `= (3*2x)/6=x` units
`=>` area of the regular hexagon `=A_h=6*sqrt3/4*x^2`
`=> A_h=6*sqrt3/4*2/sqrt3=3 " units"^2`