SOLUTION: A regular hexagon has an area of 45 sqrt3 . What is its perimeter? Please explain step by step

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Question 699054: A regular hexagon has an area of 45 sqrt3 . What is its perimeter? Please explain step by step
Answer by Positive_EV(69) About Me  (Show Source):
You can put this solution on YOUR website!
A regular hexagon can be divided into six equilateral triangles with the same area and with a side length equal to the length of the side of the hexagon. Each of these triangles will have an area of 1/6th that of the hexagon, which is %2815%2F2%29%2Asqrt%283%29.

There's a few ways to find the side of an equilateral triangle given its area, but I'll use Hero's formula. Hero's formula says that the area of a triangle with sides a, b, and c is equal to

sqrt%28s%2A%28s-a%29%2A%28s-b%29%2A%28s-c%29%29, where s = (1/2)*(a+b+c). For an equilateral triangle, a = b = c, so I will refer to the length of a side as a, and s = (1/2)(a+a+a) = (3/2)a. Hero's formula becomes:


sqrt%28%283%2F16%29%2Aa%5E4%29+=+%2815%2F2%29%2Asqrt%283%29
%283%2F16%29%2Aa%5E4+=+%28225%2F4%29%2A3
a%5E4+=+900
a+=+sqrt%2830%29

The perimeter is 6 times the length of a side, or 6%2Asqrt%2830%29.

Alternatively, you can find the side length of the triangle by using the A = (1/2)*base*height formula by drawing the height and noting that two 30-60-90 triangles are formed. The height of the triangle is going to be sqrt%283%29%2F2 times the length of a side, so you can also use

%2815%2F2%29%2Asqrt%283%29+=+%281%2F2%29%2Aa%2A%28%28sqrt%283%29%2F2%29%2Aa%29, which gives the same value of sqrt%2830%29 for a and 6%2Asqrt%2830%29 for the perimeter.