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

["# Calculating Area with Heron’s Formula: A = √[s(s−a)(s−b)(s−c)] Explained Simply", "When it comes to finding the area of a triangle when you know the lengths of all three sides, Heron’s formula is an essential mathematical tool. Whether you're a student tackling geometry or a teacher explaining triangle properties, understanding Heron’s formula is crucial. In this article, we’ll explore step-by-step how to compute the area using Heron’s formula, diving into the expression A = √[s(s−a)(s−b)(s−c)], with a practical example using the values s = 21, a = 13, b = 14, c = 15.", "---", "## What is Heron’s Formula?", "Heron’s formula provides a straightforward method to calculate the area ( A ) of a triangle given the lengths of its three sides: ( a ), ( b ), and ( c ). The key to this formula lies in the semi-perimeter ( s ), defined as:", "[\ns = \frac{a + b + c}{2}\n]", "From the semi-perimeter, Heron’s formula expresses the area as:", "[\nA = \sqrt{s(s - a)(s - b)(s - c)}\n]", "This formula works for any triangle with positive side lengths that satisfy the triangle inequality.", "---", "## Step-by-Step Calculation: How to Use Heron’s Formula", "Let’s apply Heron’s formula step-by-step using the values:\n- ( s = 21 )\n- ( a = 13 )\n- ( b = 14 )\n- ( c = 15 )", "### Step 1: Confirm Semi-Perimeter\nWe begin with the semi-perimeter:", "[\ns = \frac{13 + 14 + 15}{2} = \frac{42}{2} = 21\n]", "This matches the given ( s = 21 ), validating our input.", "### Step 2: Compute Each Part of the Product\nNext, calculate each term inside the square root:", "- ( s - a = 21 - 13 = 8 )\n- ( s - b = 21 - 14 = 7 )\n- ( s - c = 21 - 15 = 6 )", "### Step 3: Multiply the Three-Side Factors Inside the Square Root\nNow plug values into Heron’s formula:", "[\nA = \sqrt{s(s - a)(s - b)(s - c)} = \sqrt{21 \ imes 8 \ imes 7 \ imes 6}\n]", "Calculate the product step-by-step:\n- ( 21 \ imes 8 = 168 )\n- ( 168 \ imes 7 = 1176 )\n- ( 1176 \ imes 6 = 7056 )", "So:", "[\nA = \sqrt{7056}\n]", "### Step 4: Take the Square Root\nFind the square root of 7056:", "[\n\sqrt{7056} = 84\n]", "Thus, the area of the triangle is 84 square units.", "---", "## Why Heron’s Formula Matters", "Heron’s formula is valuable because it only requires knowledge of the side lengths, making it ideal when the height or angles aren’t directly known. It’s widely used in engineering, architecture, surveying, and computer graphics where precise area calculations of irregular shapes are needed.", "---", "## Summary", "- Heron’s formula: ( A = \sqrt{s(s - a)(s - b)(s - c)} )\n- Semi-perimeter: ( s = \frac{a + b + c}{2} )\n- For ( a = 13, b = 14, c = 15 ), ( s = 21 )\n- Area: ( A = \sqrt{21 \ imes 8 \ imes 7 \ imes 6} = \sqrt{7056} = 84 )", "This elegant formula demonstrates how algebraic expressions can solve geometric problems efficiently — a powerful approach every learner should master.", "---", "Keywords: Heron’s formula, A = √[s(s−a)(s−b)(s−c)], triangle area, semi-perimeter, geometry formula, formula calculation, square root of triangle area", "Optimize your geometry knowledge with Heron’s formula — easy, accurate, and universally applicable!"]