Solution: First, compute the area using Heron’s formula. Let $ a = 7 $, $ b = 10 $, $ c = 13 $. The semi-perimeter $ s $ is:

["Using Heron’s Formula to Compute the Area of a Triangle: A Step-by-Step Solution", "When working with a triangle whose side lengths are known, computing the area efficiently and accurately is essential—especially in fields like architecture, engineering, and computer graphics. One powerful method for finding a triangle’s area when all three side lengths are available is Heron’s Formula. This article guides you through the process using a concrete example: with sides $ a = 7 $, $ b = 10 $, and $ c = 13 $, we’ll compute the area using Heron’s formula, starting with the key step: calculating the semi-perimeter.", "---", "### Understanding Heron’s Formula", "Heron’s Formula allows you to compute the area of a triangle solely from the lengths of its sides. The formula is:", "$$\n\ ext{Area} = \sqrt{s(s - a)(s - b)(s - c)}\n$$", "where:\n- $ a $, $ b $, $ c $ are the lengths of the triangle’s sides,\n- $ s $ is the semi-perimeter, defined as:\n$$\ns = \frac{a + b + c}{2}\n$$", "This semi-perimeter $ s $ represents half the total perimeter and serves as the foundation for calculating the area.", "---", "### Step 1: Compute the Semi-Perimeter", "Given the side lengths $ a = 7 $, $ b = 10 $, $ c = 13 $, begin by calculating the semi-perimeter $ s $.", "Add the side lengths:", "$$\na + b + c = 7 + 10 + 13 = 30\n$$", "Now divide by 2:", "$$\ns = \frac{30}{2} = 15\n$$", "---", "### Step 2: Apply Heron’s Formula", "With $ s = 15 $, plug values into the formula:", "$$\n\ ext{Area} = \sqrt{15(15 - 7)(15 - 10)(15 - 13)}\n$$", "Simplify each term:", "$$\n= \sqrt{15 \ imes 8 \ imes 5 \ imes 2}\n$$", "Multiply inside the square root:", "$$\n= \sqrt{15 \ imes 8 \ imes 5 \ imes 2} = \sqrt{1200}\n$$", "Now simplify $ \sqrt{1200} $:", "$$\n\sqrt{1200} = \sqrt{100 \ imes 12} = 10\sqrt{12} = 10 \ imes 2\sqrt{3} = 20\sqrt{3}\n$$", "---", "### Final Result", "The area of the triangle with sides $ 7 $, $ 10 $, and $ 13 $ is $ 20\sqrt{3} $ square units.", "---", "### Why This Method Matters", "Heron’s Formula is invaluable because it requires no knowledge of angles or heights—just the three side lengths. This makes it ideal for irregular triangles commonly encountered in real-world applications. Combined with precise computation like finding the semi-perimeter first, it ensures accuracy and efficiency.", "---", "Summary:\n- Compute semi-perimeter $ s = 15 $\n- Use Heron’s formula to find area\n- Final area: $ 20\sqrt{3} $", "Mastering Heron’s Formula empowers quicker and more reliable geometric computations—key for academic success and practical problem solving."]