Solution: First, calculate the area of the triangle using Heron's formula. The semi-perimeter $ s = \frac{13 + 14 + 15}{2} = 21 $. The area is $ \sqrt{21(21-13)(21-14)(21-15)} = \sqrt{21 \cdot 8 \cdot 7 \cdot 6} = \sqrt{7056} = 84 \, \text{km}^2 $. The altitudes correspond to each side: $ h_a = \frac{2 \times 84}{13} \approx 12.92 $, $ h_b = \frac{2 \times 84}{14} = 12 $, $ h_c = \frac{2 \times 84}{15} = 11.2 $. The shortest altitude is $ \boxed{11.2} $ km.

["Solution: Calculate the Area and Shortest Altitude of a Triangle Using Heron’s Formula", "Understanding the area and altitudes of a triangle is fundamental in geometry, especially when solving problems involving area computations, optimization, or applying formulas in engineering and design. In this article, we walk through a complete solution using Heron’s formula to calculate the triangle’s area, then determine the shortest altitude among its sides.", "---", "### Step 1: Use Heron’s Formula to Find the Area", "Consider a triangle with side lengths 13 km, 14 km, and 15 km — a classic example often used to illustrate Heron’s formula.", "Step 1.1: Calculate the semi-perimeter ( s )\n[\ns = \frac{a + b + c}{2} = \frac{13 + 14 + 15}{2} = \frac{42}{2} = 21 , \ ext{km}\n]", "Step 1.2: Apply Heron’s formula\n[\n\ ext{Area} = \sqrt{s(s-a)(s-b)(s-c)}\n]\n[\n= \sqrt{21 \cdot (21-13) \cdot (21-14) \cdot (21-15)} = \sqrt{21 \cdot 8 \cdot 7 \cdot 6}\n]\n[\n= \sqrt{21 \ imes 8 \ imes 7 \ imes 6} = \sqrt{7056} = 84 , \ ext{km}^2\n]", "This step confirms the area with precision — a key foundation for further calculations.", "---", "### Step 2: Calculate Each Altitude", "The altitude corresponding to a side is derived from the relation:\n[\n\ ext{Altitude} = \frac{2 \ imes \ ext{Area}}{\ ext{Base side length}}\n]", "Since we have all three sides, compute each altitude:", "- Altitude to side 13 km:\n[\nh_a = \frac{2 \ imes 84}{13} = \frac{168}{13} \approx 12.92 , \ ext{km}\n]", "- Altitude to side 14 km:\n[\nh_b = \frac{2 \ imes 84}{14} = \frac{168}{14} = 12 , \ ext{km}\n]", "- Altitude to side 15 km:\n[\nh_c = \frac{2 \ imes 84}{15} = \frac{168}{15} = 11.2 , \ ext{km}\n]", "---", "### Step 3: Identify the Shortest Altitude", "From the computed values:\n- ( h_a \approx 12.92 , \ ext{km} )\n- ( h_b = 12 , \ ext{km} )\n- ( h_c = 11.2 , \ ext{km} )", "The shortest altitude is clearly:\n[\n\boxed{11.2} , \ ext{km}\n]", "---", "### Why This Matters", "Calculating altitudes is essential for determining how far a vertex is from the opposite side — crucial in architecture, land surveying, physics, and computer graphics. This elegant method using Heron’s formula and basic algebra gives a clear, accurate, and efficient solution.", "Whether you're solving textbook geometry problems or applying triangular analysis in real-world scenarios, mastering Heron’s formula and altitude calculation is both practical and empowering.", "---", "In summary:\nUse Heron’s formula to find the area, apply the area to compute each altitude, and compare to find the shortest. For the triangle with sides 13, 14, 15, the shortest altitude is 11.2 km — a simple yet powerful result."]