# What is the area of an equiangular triangle with perimeter 36?

Dec 6, 2015

Area= $62.35$ sq units

#### Explanation:

Perimeter = $36$

$\implies 3 a = 36$

Therefore, $a = 12$

Area of an equilateral triangle: $A = \frac{\sqrt{3} {a}^{2}}{4}$

=$\frac{\sqrt{3} \times {12}^{2}}{4}$

=$\frac{\sqrt{3} \times 144}{4}$

=$\sqrt{3} \times 36$

=$62.35$ sq units

Dec 6, 2015

$36 \sqrt{3}$

#### Explanation:

We can see that if we split an equilateral triangle in half, we are left with two congruent right triangles. Thus, one of the legs of one of the right triangles is $\frac{1}{2} s$, and the hypotenuse is $s$. We can use the Pythagorean Theorem or the properties of 30˚-60˚-90˚ triangles to determine that the height of the triangle is $\frac{\sqrt{3}}{2} s$.

If we want to determine the area of the entire triangle, we know that $A = \frac{1}{2} b h$. We also know that the base is $s$ and the height is $\frac{\sqrt{3}}{2} s$, so we can plug those in to the area equation to see the following for an equilateral triangle:

$A = \frac{1}{2} b h \implies \frac{1}{2} \left(s\right) \left(\frac{\sqrt{3}}{2} s\right) = \frac{{s}^{2} \sqrt{3}}{4}$

In your case, the perimeter of the triangle is $36$, so each side of the triangle has a side length of $12$.

$A = \frac{{12}^{2} \sqrt{3}}{4} = \frac{144 \sqrt{3}}{4} = 36 \sqrt{3}$

Nov 19, 2016

$A = 62.35$ sq units

#### Explanation:

In addition to the other answers submitted, you can do this using the trig area rule as well;

In an equilateral triangle, all the angles are 60° and all the sides are equal. IN this case as the perimeter is 36, each side is 12.

We have the 2 sides and an included angle necessary to use the area rule:

$A = \frac{1}{2} a b S \in C$

A = 1/2 xx12xx12xxSin60°

A = 6xx12xxSin60°

$A = 62.35$ sq units