Solution: First, compute the area using Heron's formula. The semi-perimeter $ s = \frac{10 + 13 + 15}{2} = 19 \, \text{cm} $. The area is $ \sqrt{19(19-10)(19-13

["Understanding Heron’s Formula: Calculating the Area of a Triangle", "When faced with a triangle whose side lengths are known but whose area is not immediately apparent, one of the most reliable mathematical tools is Heron’s formula. This elegant method allows you to compute the area using only the lengths of the three sides—no need for height or angles. In this article, we’ll walk through the step-by-step solution of Heron’s formula using an example triangle with sides 10 cm, 13 cm, and 15 cm.", "---", "### What is Heron’s Formula?", "Heron’s formula gives the area of any triangle when all three side lengths are known. The formula is:", "$$\n\ ext{Area} = \sqrt{s(s - a)(s - b)(s - c)}\n$$", "Where:\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 1: Compute the Semi-Perimeter", "For a triangle with sides 10 cm, 13 cm, and 15 cm:", "$$\ns = \frac{10 + 13 + 15}{2} = \frac{38}{2} = 19 , \ ext{cm}\n$$", "---", "### Step 2: Plug Values into Heron’s Formula", "Now apply Heron’s formula with $ s = 19 , \ ext{cm} $, $ a = 10 , \ ext{cm} $, $ b = 13 , \ ext{cm} $, $ c = 15 , \ ext{cm} $:", "$$\n\ ext{Area} = \sqrt{19(19 - 10)(19 - 13)(19 - 15)}\n$$", "$$\n= \sqrt{19 \ imes 9 \ imes 6 \ imes 4}\n$$", "---", "### Step 3: Simplify the Expression", "First compute the product inside the square root:", "$$\n19 \ imes 9 = 171 \\n171 \ imes 6 = 1026 \\n1026 \ imes 4 = 4104\n$$", "So:", "$$\n\ ext{Area} = \sqrt{4104}\n$$", "To simplify, factor 4104:\n$ 4104 = 4 \ imes 1026 = 4 \ imes 9 \ imes 114 = 4 \ imes 9 \ imes 6 \ imes 19 = 2^2 \ imes 3^2 \ imes 2 \ imes 3 \ imes 19 = 2^3 \ imes 3^3 \ imes 19 $", "$$\n\sqrt{4104} = \sqrt{2^2 \cdot 3^2 \cdot 2 \cdot 3^2 \cdot 19} = 2 \cdot 3 \cdot \sqrt{2 \cdot 3 \cdot 19} = 6\sqrt{114}\n$$", "---", "### Final Result", "The area of the triangle is:", "$$\n\ ext{Area} = 6\sqrt{114} , \ ext{cm}^2\n$$", "Approximately, since $ \sqrt{114} \approx 10.68 $, the area is about:", "$$\n6 \ imes 10.68 \approx 64.08 , \ ext{cm}^2\n$$", "---", "### Why Use Heron’s Formula?", "- No need for angle measurements or height data — ideal for field surveys or sketches.\n- Works for any triangle type, including scalene, isosceles, and obtuse.\n- Mathematically rigorous and precise, based on solid geometric principles.", "Whether you’re an architect, student, or DIY enthusiast, mastering Heron’s formula is a practical way to find triangle areas efficiently and accurately.", "Start applying Heron’s formula today — compute, analyze, and solve trianges with confidence!", "---", "Keywords: Heron’s formula, area of triangle, semi-perimeter, triangle area calculation, math tutorial, geometry, Heron’s formula example, 10-13-15 triangle, math solution steps"]