The height of this triangle would be 10.4
In order to find this, you first must find the length of the sides. Using a manipulated formula for area of an equilateral triangle, we can determine the lengths of the side. Below if the formula.
S = [tex] \frac{2}{3}3^{\frac{3}{4}} \sqrt{A} [/tex]
In this, S is the length of the side and A is the area. So we plug in and get:
S = [tex] \frac{2}{3}3^{\frac{3}{4}} \sqrt{62.4} [/tex]
S = [tex] \frac{2}{3}3^{\frac{3}{4}} 7.89 [/tex]
S = 12
Now that we have the side as 12, we can use the Pythagorean Theorem to find the height. If you split a equilateral triangle down the middle, you are left with two right triangles. Using this right triangle, the hypotenuse would be 12, the first leg would be 6 (half of the base) and the height would be the other leg. So we plug in and solve.
[tex] a^{2} + b^{2} = c^{2} [/tex]
[tex] 6^{2} + h^{2} = 12^{2} [/tex]
[tex] 36 + h^{2} = 144 [/tex]
[tex] h^{2} = 108 [/tex]
h = 10.4
