# What is the area of a equilateral triangle with a side of 12?

Dec 9, 2014

The formula for the Area of an equilateral triangle with side s is $A = \frac{{s}^{2} \sqrt{3}}{4}$

For an equilateral triangle, the sides are equal and the angles are equal. So each angle is 60 degrees. If we are to drop a vertical line from the vertex angle we divide the opposite side in two equal parts. The vertex angle is also equally divided into two. Thus, we form a 30°-60°-90° triangle.

The height , h, of the right triangle in terms of the sides (s) of the equilateral triangle is $\frac{s \sqrt{3}}{2}$

This height is also the height of the equilateral triangle whose base is s.

Generally the Area of any triangle is,

$A = \frac{\left(b a s e\right) \cdot \left(h e i g h t\right)}{2}$

plugging in the values,

$A = \frac{\left(s\right) \cdot \frac{s \sqrt{3}}{2}}{2}$

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

$s = 12$

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

$A = 36 \sqrt{3}$ square units