\[ c = \sqrt{a^2 + b^2} = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \text{ meters} \]

["Understanding the Pythagorean Theorem: How to Calculate the Hypotenuse", "The mathematical secret behind right triangles has fascinated scientists, engineers, and students for centuries. At the heart of this lies a simple yet powerful equation:\n[ c = \sqrt{a^2 + b^2} ]\nwhere ( c ) represents the hypotenuse—the longest side opposite the right angle—while ( a ) and ( b ) are the two shorter perpendicular sides.", "### What Is the Hypotenuse and Why Does It Matter?", "The hypotenuse (( c )) is fundamental in geometry, trigonometry, and real-world applications like construction, navigation, and physics. Its calculation relies on the Pythagorean Theorem, discovered by the ancient Greek mathematician Pythagoras. This theorem states that:", "> In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.", "That translates directly to the formula:\n[ c^2 = a^2 + b^2 ]\nor rearranged:\n[ c = \sqrt{a^2 + b^2} ]", "### Step-by-Step Example: Calculating the Hypotenuse", "Let’s apply the formula with a classic example:\n[ c = \sqrt{5^2 + 12^2} ]", "1. Square each leg:\n ( 5^2 = 25 )\n ( 12^2 = 144 )", "2. Add the squares:\n ( 25 + 144 = 169 )", "3. Take the square root:\n ( \sqrt{169} = 13 )", "So, the hypotenuse measures 13 meters, demonstrating how the theorem efficiently connects side lengths in a right triangle.", "### Real-Life Applications of the Hypotenuse Formula", "Knowing how to calculate the hypotenuse helps in:", "- Determining diagonal distances (e.g., across a rectangular room)\n- Verifying structural integrity in building design\n- Navigating via trigonometric calculations\n- Solving physics problems involving vectors and forces", "### Mastering the Pythagorean Theorem", "Beyond memorizing the formula, understanding the logic behind it strengthens spatial reasoning and problem-solving skills. Whether you’re drawing triangles on paper or writing code for calculations, mastering ( c = \sqrt{a^2 + b^2} ) remains a vital building block in mathematics.", "---", "Remember:\nHave you calculated a hypotenuse recently? Practice with different values to strengthen your grasp of this essential trigonometric principle!", "#PythagoreanTheorem #HypotenuseCalculation #TriangleMath #GeometryEducation #LearningGeometry #MathFormulas", "---", "Keywords: Pythagorean theorem, hypotenuse calculation, ( c = \sqrt{a^2 + b^2} ), right triangle math, geometry tutorial, diagonal distance formula, trigonometric basics", "Meta Description: Learn how to calculate the hypotenuse using ( c = \sqrt{a^2 + b^2} ) with a clear example, ( 5^2 + 12^2 = 169 ), resulting in ( c = 13 ) meters—perfect for mastering right triangle geometry."]