To find the area of the triangle with sides \(a = 13\), \(b = 14\), and \(c = 15\), we use Heron's formula. The semi-perimeter \(s\) is:

["# Find the Area of a Triangle Using Heron’s Formula: A Step-by-Step Guide", "Calculating the area of a triangle when all three side lengths are known can be efficiently done using Heron’s Formula. Whether for math students, engineers, or simply curious learners, understanding this formula helps solve real-world geometry problems. In this article, we’ll explore how to find the area of a triangle with sides (a = 13), (b = 14), and (c = 15), focusing on calculating the semi-perimeter (s), a crucial first step.", "---", "## What is Heron’s Formula?", "Heron’s formula allows you to compute the area (A) of any triangle from its side lengths (a), (b), and (c) without needing angles or heights. The formula is:", "[\nA = \sqrt{s(s - a)(s - b)(s - c)}\n]", "where (s) is the semi-perimeter of the triangle.", "---", "## Step 1: Compute the Semi-Perimeter (s)", "The semi-perimeter (s) is found by averaging the triangle’s perimeter. Since perimeter (P = a + b + c), then:", "[\ns = \frac{a + b + c}{2}\n]", "For our triangle with sides (a = 13), (b = 14), and (c = 15):", "[\ns = \frac{13 + 14 + 15}{2} = \frac{42}{2} = 21\n]", "✅ Semi-perimeter (s = 21)", "This value is essential because it standardizes the values inside the square root in Heron’s formula, ensuring accuracy.", "---", "## Why is the Semi-Perimeter Important?", "The semi-perimeter serves as a normalized measure that balances the three side lengths. It simplifies computations and helps avoid large intermediate numbers, especially when applying the square root. Without (s), applying Heron’s formula directly to side lengths becomes unwieldy.", "---", "## Using the Semi-Perimeter to Find Area", "Now that we know (s = 21), (a = 13), (b = 14), and (c = 15), plug into Heron’s formula:", "[\nA = \sqrt{21(21 - 13)(21 - 14)(21 - 15)} = \sqrt{21 \ imes 8 \ imes 7 \ imes 6}\n]", "Calculate step-by-step:", "- (21 \ imes 8 = 168)\n- (7 \ imes 6 = 42)\n- (168 \ imes 42 = 7056)", "[\nA = \sqrt{7056} = 84\n]", "So, the area of the triangle is 84 square units.", "---", "## Summary", "- The semi-perimeter (s) for triangle sides 13, 14, and 15 is (s = 21).\n- This value is found by (s = \frac{a + b + c}{2}).\n- Heron’s formula (A = \sqrt{s(s - a)(s - b)(s - c)}) efficiently computes the triangle’s area.\n- For sides 13, 14, 15, the area is 84 — a classic example often used in geometry.", "---", "## Don’t Miss These Next Steps", "- Practice with other triangles to master Heron’s formula.\n- Learn how this formula applies to real-world applications like land surveying and construction.\n- Compare Heron’s formula with vector-based area methods for advanced geometry.", "Try computing the area of any triangle using Heron’s method — start with (s = \frac{a + b + c}{2}) and you’ll unlock powerful geometry tools!", "---", "Keywords: triangle area, Heron’s formula, semi-perimeter (s), calculate triangle area, area of triangle 13-14-15, geometry formula, semi-perimeter calculation, Euclidean geometry."]