# What is the formula for the area of an equilateral triangle?

#### Understand the Problem

The question is asking for the mathematical formula that calculates the area of an equilateral triangle, which is a triangle with all three sides of equal length.

\frac{\sqrt{3}}{4}a^2

The final formula for the area of an equilateral triangle with side length $a$ is $\frac{\sqrt{3}}{4}a^2$

#### Steps to Solve

1. Identify the formula for the area of an equilateral triangle

For an equilateral triangle with side length $a$, the formula for the area $A$ is derived using the formula for the area of a general triangle and some geometric properties.

2. Use the formula for the area of a general triangle

The area $A$ of a triangle is given by

$$A = \frac{1}{2} \times base \times height$$

For an equilateral triangle, all sides are equal and the height can be calculated using the Pythagorean theorem.

3. Calculate the height of the equilateral triangle

Split the equilateral triangle into two right-angled triangles. Each of these right triangles has a base of $\frac{a}{2}$ and a hypotenuse of $a$. Use the Pythagorean theorem to find the height $h$:

$$a^2 = \left(\frac{a}{2}\right)^2 + h^2$$

Solving for $h$,

$$h^2 = a^2 - \left(\frac{a}{2}\right)^2$$

$$h^2 = a^2 - \frac{a^2}{4}$$

$$h^2 = \frac{3a^2}{4}$$

$$h = \frac{\sqrt{3}}{2}a$$

4. Substitute the height into the area formula

Substitute the height calculated into the general area formula to get the area of the equilateral triangle:

$$A = \frac{1}{2} \times a \times \frac{\sqrt{3}}{2}a$$

Simplifying,

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

The final formula for the area of an equilateral triangle with side length $a$ is $\frac{\sqrt{3}}{4}a^2$