HELP ASAP PLEASE!

Area of an equilateral triangle  =  

(1/2) s^2 ( √ 3 / 2 )  =

( √ 3 / 4 )  s^2            where s is a side length

Bur....since a side length is numerically equal to this  area  and we have three equal sides....the perimeter =

3 * ( √ 3 / 4 )  s^2  =

( 3 √ 3 / 4 )  s^2

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.

Let h = height of the equilateral triangle.

Let a = one of its sides.

1. Pythagoras: h=?

\(\begin{array}{|rcll|} \hline \left( \frac{a}{2} \right)^2 + h^2 &=& a^2 \\ \frac{a^2}{4} + h^2 &=& a^2 \quad & | \quad -\frac{a^2}{4} \\ h^2 &=& a^2 - \frac{a^2}{4} \\ h^2 &=& \frac34 a^2 \\ \mathbf{ h } & \mathbf{=} & \mathbf{ \frac{a}{2}\sqrt{3} } \\ \hline \end{array}\)

2. Area A of the equilateral triangle:

\(\begin{array}{|rcll|} \hline A &=& \frac{a\cdot h}{2} \quad & | \quad h = \frac{a}{2}\sqrt{3} \\ &=& \frac{a}{2} \cdot \frac{a}{2}\sqrt{3} \\ \mathbf{ A } & \mathbf{=} & \mathbf{ \frac{a^2}{4}\sqrt{3} } \\ \hline \end{array}\)

3. Area = a

\(\begin{array}{|rcll|} \hline A &=& a \\ \frac{a^2}{4}\sqrt{3} &=& a \\ \frac{a}{4}\sqrt{3} &=& 1 \\ a &=& \frac{4}{ \sqrt{3} } \cdot \frac{\sqrt{3}} {\sqrt{3}} \\ \mathbf{ a } & \mathbf{=} & \mathbf{ \frac{4}{3} \sqrt{3} }\\ \hline \end{array}\)

4. Perimeter = 3a

\(\begin{array}{|rcll|} \hline \text{Perimeter} &=& 3\cdot a \quad & | \quad a = \frac{4}{3} \sqrt{3} \\ &=& 3\cdot \frac{4}{3} \sqrt{3} \\ \mathbf{ \text{Perimeter} } & \mathbf{=} & \mathbf{ 4\cdot \sqrt{3} } \\ \hline \end{array}\)

Using Heron's formula the area of an equilateral triangle can be written as:

A = √(s(s-a)^3)  where a is side length and s is semi-perimeter (=3a/2 here)

If A = a numerically. then:

a = √(3a/2(a/2)^3)   or   a = (a/2)^2*√3  so   1 = (a/4)√3    or   a = 4/√3 → (4√3)/3

Hence perimeter = 3a → 4√3

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