\[ \text{Área} = \sqrt{24(24-10)(24-17)(24-21)} = \sqrt{24 \times 14 \times 7 \times 3} = \sqrt{7056} = 84 \, \text{km}^2 \]

["# Solving an Irrational Square Root: A Step-by-Step Breakdown of Area Calculation", "Area, geometry, and algebra often meet in solving real-world problems — and sometimes the solution involves an unexpected square root. Today, we explore a classic example: calculating the area using Heron’s formula applied to a geometric shape defined by [ \ ext{Área} = \sqrt{24(24-10)(24-17)(24-21)} = \sqrt{7056} = 84 , \ ext{km}^2 ]. Let’s dive into how this elegant expression simplifies to a whole number.", "---", "## Understanding the Formula: Heron’s Formula and Square Roots", "The formula [ \ ext{Área} = \sqrt{s(s-a)(s-b)(s-c)} ] is Heron’s formula, used to calculate the area of a triangle when all three side lengths ( a, b, c ) are known, with ( s = \frac{a+b+c}{2} ) being the semi-perimeter.", "In our case:\n- ( a = 24 , \ ext{km} )\n- ( b = 14 , \ ext{km} )\n- ( c = 7 , \ ext{km} )", "Plugging these values in, we compute the semi-perimeter first:\n[\ns = \frac{24 + 14 + 7}{2} = \frac{45}{2} = 22.5 , \ ext{km}\n]\nThis semi-perimeter may seem unusual — but it’s key to unlocking the area.", "---", "## Breaking Down the Expression", "Now substitute into Heron’s formula:\n[\n\ ext{Área} = \sqrt{24 \ imes (24 - 10) \ imes (24 - 17) \ imes (24 - 21)}\n]\nSimplify step-by-step:\n- ( 24 - 10 = 14 )\n- ( 24 - 17 = 7 )\n- ( 24 - 21 = 3 )", "So the expression becomes:\n[\n\sqrt{24 \ imes 14 \ imes 7 \ imes 3}\n]", "Next, multiply the numbers inside the square root:\n[\n24 \ imes 14 = 336\n]\n[\n7 \ imes 3 = 21\n]\n[\n336 \ imes 21 = 7056\n]", "Thus:\n[\n\ ext{Área} = \sqrt{7056} = 84 , \ ext{km}^2\n]", "---", "## Why the Result is a Nice Whole Number", "Heron’s formula often yields perfect squares when the side lengths are chosen carefully — a hidden symmetry rooted in number theory. In this case, the product inside the square root forms a composite number whose square root simplifies cleanly due to perfectly balanced factor groupings.", "The expression ( 24 \ imes 14 \ imes 7 \ imes 3 ) factors neatly into known perfect squares:\n- ( 24 = 4 \ imes 6 )\n- ( 14 = 2 \ imes 7 )\n- ( 7 = 7 )\n- ( 3 = 3 )", "Rearranging:\n[\n\sqrt{(4 \ imes 6) \ imes (2 \ imes 7) \ imes 7 \ imes 3} = \sqrt{(4 \ imes 4) \ imes (3 \ imes 3) \ imes (7 \ imes 7)} \ imes \sqrt{2 \ imes 6}\n]\nBut more simply, grouping properly:\n[\n= \sqrt{(24 \ imes 7) \ imes (14 \ imes 3)} = \sqrt{168 \ imes 42} = \sqrt{7056}\n]", "And ( \sqrt{7056} = 84 ), since ( 84^2 = 7056 ). This confirms the area is exactly 84 km², a clean and useful result.", "---", "## Practical Applications in Real Life", "This calculation isn’t just academic. Heron’s formula, and problems like this, are applied in:\n- Surveying land: When measuring irregular plots bordered by straight segments.\n- Urban planning: Calculating green spaces or irregularly shaped zones.\n- Engineering: Structural design involving triangular components.", "The fact that the area simplifies to a whole number means precision in measurement and calculation — essential for accurate land use and construction.", "---", "## Final Thoughts", "While square roots can seem mysterious, problems like [ \ ext{Área} = \sqrt{24(24-10)(24-17)(24-21)} ] showcase how structured numbers and careful algebraic manipulation yield clean, practical results. This example reminds us that mathematics often balances complexity with elegance — and sometimes reveals perfect squares through clever grouping.", "Whether for homework, real-world applications, or simply expanding your mathematical toolkit, mastering Heron’s formula helps unlock deeper geometric understanding.", "---", "### Quick Recap:\n- Values: ( a=24,\ ext{km}, b=14,\ ext{km}, c=7,\ ext{km} )\n- Semi-perimeter: ( s = 22.5,\ ext{km} )\n- Area: ( \sqrt{24 \cdot 14 \cdot 7 \cdot 3} = \sqrt{7056} = 84,\ ext{km}^2 )", "Ready to calculate geometric areas with confidence? This one’s a smooth 84 — no irrationality needed.", "---", "Keywords: Area calculation, Heron’s formula, square root evaluation, 84 km², geometric formulas, algebraic simplification, land surveying math, perfect square under root, intuitive geometry."]