1. **State the problem:** We need to find the area of an equilateral triangle with a perimeter of 36 cm using Heron's formula. 2. **Find the side length:** Since the triangle is equilateral, all sides are equal. The perimeter $P$ is given by: $$P = a + b + c = 3a$$ Given $P = 36$ cm, solve for $a$: $$3a = 36$$ $$a = \frac{36}{3} = 12 \text{ cm}$$ 3. **Apply Heron's formula:** The semi-perimeter $s$ is half the perimeter: $$s = \frac{P}{2} = \frac{36}{2} = 18$$ Heron's formula for the area $A$ is: $$A = \sqrt{s(s - a)(s - b)(s - c)}$$ Since $a = b = c = 12$, substitute: $$A = \sqrt{18(18 - 12)(18 - 12)(18 - 12)} = \sqrt{18 \times 6 \times 6 \times 6}$$ 4. **Calculate the area:** $$A = \sqrt{18 \times 6^3} = \sqrt{18 \times 216} = \sqrt{3888}$$ 5. **Simplify the square root:** $$3888 = 36 \times 108 = 36 \times (36 \times 3) = 36^2 \times 3$$ So, $$A = \sqrt{36^2 \times 3} = 36 \sqrt{3}$$ 6. **Approximate the value:** $$\sqrt{3} \approx 1.732$$ $$A \approx 36 \times 1.732 = 62.352$$ 7. **Round to the nearest square centimeter:** $$A \approx 62 \text{ square centimeters}$$ **Final answer:** The area of the equilateral triangle is approximately 62 square centimeters.