To calculate the area of an equilateral triangle, you can use the formula:

An equilateral triangle is characterized by having all three sides of equal length and all three angles measuring $60$ degrees. The formula for the area of an equilateral triangle is derived from the standard formula for the area of a triangle, which is given by:

In the case of an equilateral triangle, the height can be determined using Pythagoras’ theorem.

To derive the area formula, consider an equilateral triangle with each side measuring a length of $a$. If you draw a perpendicular line from one vertex to the midpoint of the opposite side, this line will bisect that side into two equal segments, each measuring $\frac{a}{2}$. This perpendicular line represents the height ($h$) of the triangle.

Applying Pythagoras’ theorem to one of the resulting right triangles, we have:

Substituting in the values gives:

Rearranging the equation yields:

This simplifies to:

Taking the square root of both sides, we find:

Now, we can substitute this height back into the area formula for a triangle:

Substituting the base ($a$) and the height ($\frac{\sqrt{3}}{2} a$), we have:

This simplifies to:

Thus, the area of an equilateral triangle with a side length of $a$ is given by:

This formula is particularly useful as it allows you to compute the area directly from the side length without the need to separately calculate the height.