The area of an equilateral triangle is numerically equal to the length of one of its sides. What is the perimeter of the triangle, in units? Express your answer in simplest radical form.
The formula for the area of an equilateral triangle is $\frac{s^2 \sqrt{3}}{4}$. This must equal $s$. Setting the two equal and solving, we get \begin{align*}
s&=\frac{s^2 \sqrt{3}}{4} \\
4s&=s^2\sqrt{3} \\
4 &= s\sqrt{3} \\
4\sqrt{3}&= 3s \\
\frac{4\sqrt{3}}{3} &=s
\end{align*} Thus, the perimeter of the triangle is $3s=\frac{4\sqrt{3}}{3} \cdot 3 = \boxed{4\sqrt{3}} \text{units}$.
4\sqrt{3}
Level 5
Geometry
Problem: The area of an equilateral triangle is numerically equal to the length of one of its sides. What is the perimeter of the triangle, in units? Express your answer in simplest radical form.
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the perimeter of the equilateral triangle, let the length of one side be denoted as s. The area A of an equilateral triangle can be calculated using the formula A = (sqrt(3)/4) * s^2. According to the problem, this area is equal to the length of one of its sides, so we set up the equation:

(sqrt(3)/4) * s^2 = s.

To solve for s, we first eliminate s from both sides, assuming s is not zero:

(sqrt(3)/4) * s = 1.

Next, we multiply both sides by 4 to isolate s:

sqrt(3) * s = 4.

Now, we divide both sides by sqrt(3):

s = 4/sqrt(3).

To express s in a simpler form, we rationalize the denominator:

s = (4/sqrt(3)) * (sqrt(3)/sqrt(3)) = (4sqrt(3))/3.

Now that we have the length of one side, we can find the perimeter P of the equilateral triangle, which is given by the formula P = 3s. Substituting our expression for s:

P = 3 * (4sqrt(3)/3).

The 3s cancel out:

P = 4sqrt(3).

Thus, the perimeter of the triangle is 4sqrt(3).<|END_OF_LATENT|> The final answer is $4\sqrt{3}$.