The area of an equilateral triangle can be expressed with respect to the length of one of its sides. Let’s denote the length of a side of the equilateral triangle as x.
To find the area, we can use the formula for the area of an equilateral triangle:
Area = (sqrt(3) / 4) * x²
Here, sqrt(3) is the square root of 3, which is approximately 1.732. This formula arises from the general formula for the area of a triangle (1/2 * base * height), adapted specifically for equilateral triangles where all sides and angles are equal.
To derive this, we note that in an equilateral triangle, the height can be calculated using the Pythagorean theorem. When a perpendicular is drawn from one vertex to the opposite side, it divides the triangle into two right triangles. The height h can be expressed as:
h = sqrt(x² – (x/2)²) = sqrt(x² – x²/4) = sqrt(3x²/4) = (sqrt(3)/2)*x
Plugging the height back into the area formula gives:
Area = (1/2) * base * height = (1/2) * x * (sqrt(3)/2)*x = (sqrt(3)/4) * x².
Thus, the area of an equilateral triangle as a function of the side length x is:
Area(x) = (sqrt(3) / 4) * x²