If the area of equilateral triangle is $3\sqrt3$ cm$^2$ , then what is the height of the equilateral triangle?

Question

If the area of equilateral triangle is $3\sqrt3$ cm$^2$ , then what is the height of the equilateral triangle?

I am stuck with this question
I solved it like this:
Area of equilateral triangle is $\frac{a\sqrt3}4$
So, $\frac{\sqrt3}4 \cdot a = 3 \sqrt 3$
$a = \frac{3\sqrt3}{\sqrt3/4} = 3\sqrt3 \cdot \frac4{\sqrt3}$; $\sqrt 3$'s cancel
$a = 3 \cdot 4 = 12$
I found the side is $12$.
How do I find the height with the side length?
And also kindly say if I made any mistake in my calculations.

Answer 1

The area of an equilateral triangle is (root3 a^2)/4

The height of an equilateral triangle is (root3 a)/2

So

3root3 = a^2 root3/4

=> 3 = a^2/4
=> a = root(12) cm
Hence height is :
root3 * root(12) * 1/2 = 3 cm

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Answer 2

Consider

Why is the height $\frac{\sqrt{3}}{2}s$? So what is the area?

Answer 3

The reason the formula for the area of an equilateral triangle is $\frac {\sqrt{3}a^2}4$ (you didn't write it down correctly) is because if you take an equilateral triangle with sides $a$ and cut it into two right triangles the side of the right triangle will be a hypotenuse $a$, the original side of the equilateral triangle, a leg of $\frac 12 a$ but half a side of the triangle. The third side is the height of the triangle, and if we calll it $h$ this is a right triangle so we must have $h^2 + (\frac 12 a)^2 = a^2$ by the Pythagorean theorem.

So Solving for $h$ we heve $h =\frac{\sqrt 3 a}2$. So the area of a triangle is $\frac 12 b\cdot h$ and $b = a$ and $h =\frac {\sqrt 3 a}2$ so the area of the triangle is $\frac {\sqrt 3 a^2}4$.

But in comming up with a formula you had to figure out the height was $h = \frac {\sqrt 3a}2$.

So we have $\frac {\sqrt 3a^2}4 = 3\sqrt 3$ and we have to figure out what $\frac {3a}2$ is. Solve for $a$ and plug it in.