The Pythagorean Theorem Explained
The Pythagorean theorem is one of the oldest and most widely applicable theorems in mathematics. It states that in any right triangle, the square of the hypotenuse (the longest side, opposite the right angle) equals the sum of the squares of the other two sides. It is used in construction, navigation, computer graphics, physics, and countless other fields.
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Formula
Pythagorean Theorem
Solve for any side of a right triangle using the Pythagorean theorem.
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Worked Example
Given:
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FAQs
How do I find a leg of the triangle if I know the hypotenuse?
Rearrange the formula: a = √(c² - b²). For example, if the hypotenuse is 10 and one leg is 6: missing leg = √(10² - 6²) = √(100 - 36) = √64 = 8.
What are Pythagorean triples?
Pythagorean triples are sets of three integers that satisfy the Pythagorean theorem exactly. Common examples: (3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25). Multiples of these also work: (6, 8, 10), (9, 12, 15). They are useful in construction because they guarantee a perfect right angle.
Does the Pythagorean theorem work in 3D?
Yes — the 3D extension is: d = √(x² + y² + z²) where d is the diagonal distance between two points in three-dimensional space. This formula is used in computer graphics, 3D modelling, and physics to calculate distances between points in space.