Find the area of an equilateral triangle of side “a” using Heron's formula.

To find the area of an equilateral triangle using Heron's formula, we first need to understand the properties of the triangle and the formula itself.

Understanding the Triangle

An equilateral triangle has all three sides equal. If we denote the length of each side as “a,” then the semi-perimeter (s) can be calculated as:

• s = (a + a + a) / 2 = 3a / 2

Applying Heron's Formula

Heron's formula for the area (A) of a triangle is given by:

• A = √(s(s - a)(s - b)(s - c))

In the case of an equilateral triangle, since all sides are equal (a = b = c), we can substitute:

• A = √(s(s - a)(s - a)(s - a))

Calculating the Area

Now substituting the value of s:

• A = √((3a/2)((3a/2) - a)((3a/2) - a)((3a/2) - a))

Calculating further:

• A = √((3a/2)(a/2)(a/2)(a/2))

This simplifies to:

• A = √((3a/2) * (a/2)^3) = √((3a^4) / 16)

Finally, the area can be expressed as:

• A = (√3 / 4) * a²

Final Result

Thus, the area of an equilateral triangle with side length “a” is:

A = (√3 / 4) * a²