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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.

MeepMeep  Jul 9, 2017
#1
+87656
+1

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

CPhill  Jul 10, 2017
#2
+19841
+1

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?

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

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

heureka  Jul 10, 2017
edited by heureka  Jul 10, 2017
#3
+26838
+1

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

Alan  Jul 10, 2017