Question 699054
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 {{{(15/2)*sqrt(3)}}}.<br>

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<br>

{{{sqrt(s*(s-a)*(s-b)*(s-c))}}}, 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:<br>

{{{sqrt((3/2)*a*((1/2)*a)*((1/2)*a)*((1/2)*a)) = (15/2)*sqrt(3)}}}
{{{sqrt((3/16)*a^4) = (15/2)*sqrt(3)}}}
{{{(3/16)*a^4 = (225/4)*3}}}
{{{a^4 = 900}}}
{{{a = sqrt(30)}}}<br>

The perimeter is 6 times the length of a side, or {{{6*sqrt(30)}}}.<br>

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(3)/2}}} times the length of a side, so you can also use<br>

{{{(15/2)*sqrt(3) = (1/2)*a*((sqrt(3)/2)*a)}}}, which gives the same value of {{{sqrt(30)}}} for a and {{{6*sqrt(30)}}} for the perimeter.