Short answer 150 sqrt(3)
Remark
A regular hexagon can be divided into 6 equilateral triangles (all sides are equal of these tringles). The next thing to do is find 1/2 the base using the 30 60 90 triangle that the perpendicular bisector forms with the base of one of these triangles.
Step One
Find 1/2 the base of one of the sides of the equilateral triangles.
The ratio of 1/2 the base (b1) to the height of the triangle sqrt(3) / 3
Givens
b1 = ??
h = 5sqrt(3)
ratio of b1 /h = sqrt(3)/3
b1 / 5*sqrt(3) = sqrt(3) / 3 Cross multiply
3*b1 = 5*sqrt(3) * sqrt(3)
3*b1 = 5*3 Divide both sides by 3
b1 = 5
Remember that is half the base.
Step two
Find the length of the full base of one of the 6 triangles.
1/2 base = 5
base = 10
Step Three
Find the area of 1 of the triangles.
b = 10
h = 5sqrt(3)
Area = 1/2 b*h
Area = 1/2 * 10 * 5 sqrt(3)
Area = 25 sqrt(3)
Step Four
Find the area of all 6 triangles.
1 triangle has an area of 25*sqrt(3)
6 triangles have an area of 6 *25sqrt(3) = 150 sqrt(3)