A = \sqrt{s(s - a)(s - b)(s - c)} = \sqrt{21(21 - 13)(21 - 14)(21 - 15)} = \sqrt{21 \cdot 8 \cdot 7 \cdot 6}

["# Understanding Heron’s Formula: Calculating the Area of a Triangle Using Its Side Lengths", "When it comes to finding the area of a triangle given only the lengths of its three sides, Heron’s formula offers a powerful and elegant solution. At the heart of this formula lies the expression A = √[s(s − a)(s − b)(s − c)], a mathematical gem that leverages the semi-perimeter of the triangle to compute its area without requiring height or angles.", "## What is Heron’s Formula?", "Heron’s formula allows you to calculate the area of any triangle—whether acute, obtuse, or right—when you know the lengths of all three sides: a, b, and c. The formula is named after the ancient Greek mathematician Heron of Alexandria, who documented it around the 1st century CE.", "The formula states:", "[\nA = \sqrt{s(s - a)(s - b)(s - c)}\n]", "where:\n- A is the area of the triangle,\n- a, b, and c are the lengths of the triangle’s sides,\n- s is the semi-perimeter, calculated as ( s = \frac{a + b + c}{2} ).", "## Step-by-Step Calculation Using Heron’s Formula", "Let’s apply Heron’s formula step-by-step using a classic example:", "Given side lengths:\na = 21, b = 14, c = 15", "### Step 1: Compute the semi-perimeter\n[\ns = \frac{21 + 14 + 15}{2} = \frac{50}{2} = 25\n]", "### Step 2: Apply Heron’s formula\n[\nA = \sqrt{s(s - a)(s - b)(s - c)} = \sqrt{21(21 - 13)(21 - 14)(21 - 15)}\n]", "Calculating each term inside the square root:\n- ( s - a = 25 - 21 = 4 )\n- ( s - b = 25 - 14 = 11 )\n- ( s - c = 25 - 15 = 10 )", "Now substitute:\n[\nA = \sqrt{21 \cdot 4 \cdot 11 \cdot 10}\n]", "### Step 3: Simplify the product\nCalculate the product inside the root:\n[\n21 \ imes 4 = 84\n]\n[\n84 \ imes 11 = 924\n]\n[\n924 \ imes 10 = 9240\n]", "So,\n[\nA = \sqrt{9240}\n]", "### Step 4: Simplify √9240\nFactor 9240 to simplify the square root:\n[\n9240 = 2^3 \cdot 3 \cdot 5 \cdot 7 \cdot 11\n]", "Group perfect squares:\n[\n\sqrt{9240} = \sqrt{4 \cdot 2310} = 2\sqrt{2310}\n]", "But 2310 has no perfect square factors beyond 4, so this is fully simplified. However, direct computation gives:\n[\n\sqrt{9240} \approx 96.12 \quad (\ ext{using a calculator})\n]", "> Note: While √9240 doesn’t simplify to a whole number, Heron’s formula ensures accuracy even with non-integer areas.", "## Why Use Heron’s Formula?", "- No need for height or angles: Ideal when only side measurements are available.\n- Accurate for all triangle types: Whether equilateral, isosceles, or scalene.\n- Foundation for advanced geometry: Used in surveying, architecture, and computer graphics.", "## Real-World Applications", "Heron’s formula is widely applied in:\n- Civil engineering for land boundary calculations\n- Geography for area estimation from satellite-derived side measurements\n- Physics and trigonometry for area-based derivations", "## Conclusion", "Heron’s formula delivers a precise and straightforward method to compute the area of any triangle from its side lengths. Whether solving textbook problems or tackling real-world measurement challenges, understanding and applying:", "[\nA = \sqrt{s(s - a)(s - b)(s - c)}\n]", "is essential for students, engineers, and mathematicians alike. With side lengths a = 21, b = 14, and c = 15, we find the area is √9240, demonstrating the formula’s practicality and elegance.", "---", "Keywords for SEO: Heron’s formula, area of a triangle, semi-perimeter formula, √[s(s−a)(s−b)(s−c)], triangle area calculation, side lengths to area, classic geometry formula."]