What is the area formula for an equilateral triangle?

The area formula for an equilateral triangle is derived from its unique properties. An equilateral triangle has all three sides of equal length, denoted by (s).

To find the area, you can use the general triangle area formula (A = \frac{1}{2} \times \text{base} \times \text{height}). In the case of an equilateral triangle, the height can be calculated using the Pythagorean theorem. The height divides the triangle into two 30-60-90 right triangles. The height (h) corresponds to the side length (s) as follows:

1. The altitude (height) forms a right triangle with the base (which is half of (s), or (\frac{s}{2})) and the hypotenuse (which is (s)).

2. Using the properties of a 30-60-90 triangle, the height can be calculated as (h = \frac{\sqrt{3}}{2}s).

Substituting this height back into the area formula gives:

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A = \frac{1}{2} \times s \times \frac{\sqrt{3}}{2}s = \