The area of an equilateral triangle is √3. What is the perimeter of the triangle?

Equilateral Triangle Area to Perimeter Calculation

This solution explains how to find the perimeter of an equilateral triangle when its area is given as $\sqrt{3}$. We will use the standard formulas for the area and perimeter of an equilateral triangle.

Understanding Equilateral Triangles

An equilateral triangle is a triangle where all three sides are equal in length. Consequently, all three internal angles are also equal, each measuring 60 degrees. Let's denote the length of a side of the equilateral triangle as '$s$ '.

Area Formula and Calculation

The formula for the area of an equilateral triangle is given by:

`Area = $\frac{\sqrt{3}}{4}{s}^2$ `

We are given that the area of the triangle is $\sqrt{3}$. We can set up the equation:

`$\frac{\sqrt{3}}{4}{s}^2=\sqrt{3}$ `

To find the side length '$s$ ', we can solve this equation:

- Divide both sides by $\sqrt{3}$: `$\frac{1}{4}{s}^2=1$ `
- Multiply both sides by 4: `$s^2=4$ `
- Take the square root of both sides: `$s=\sqrt{4}$ `
- Since the side length must be positive: `$s=2$ `

So, the length of each side of the equilateral triangle is 2 units.

Perimeter Formula and Calculation

The perimeter of a triangle is the sum of the lengths of its three sides. For an equilateral triangle, since all sides are equal (length '$s$ '), the formula is:

`Perimeter = $s+s+s=3s$ `

Now, substitute the value of '$s$ ' we found (which is 2) into the perimeter formula:

`Perimeter = $3\times 2$ `

Perimeter = 6

Conclusion

Therefore, the perimeter of the equilateral triangle with an area of $\sqrt{3}$ is 6 units.