Final answer:
The area of an equilateral triangle with side 'a' is calculated using Heron's formula. The semi-perimeter s is half the triangle's perimeter (3a/2), and the area is found by A = (a^2√3)/4.
Step-by-step explanation:
Calculating the Area of an Equilateral Triangle Using Heron's Formula
To calculate the area of an equilateral triangle with side 'a' using Heron's formula, you will first need to find the semi-perimeter of the triangle (s). The semi-perimeter is half of the triangle's perimeter, and since all sides are equal in an equilateral triangle, this is simply s = 3a/2. Once you have s, you can plug it into Heron's formula to find the area (A), which is A = √[s(s-a)(s-a)(s-a)]. In the case of an equilateral triangle, this simplifies to A = √[s^3(s-a)], since s-a is the same for all three sides. Expressing s and substituting it into the formula, you get A = √[(3a/2)^3((3a/2)-a)], further simplifying to A = (a^2√3)/4, as the area of an equilateral triangle is a well-known derivative of this formula.
An example would be if the side of the triangle is 60 cm long. Using the formula, we find that the area is (60^2√3)/4 square centimeters.