The Pythagorean Theorem: Beyond the Right-Angled Triangle at School
thecalculator.tech
19 June 2026 · 4 min read
Discover why the Pythagorean theorem is much more than a geometry exercise: architecture, GPS, screens and how it is used in real-world problems.
The theorem and its formula
In any right-angled triangle: a² + b² = c², where c is the hypotenuse (the longest side, opposite the right angle) and a, b are the legs. It has more than 370 known proofs — more than any other mathematical theorem. The most visual: build squares on each side; the area of the large one equals the sum of the two small ones.
Applications in architecture and construction
Builders use the theorem to verify that an angle is right without a protractor: if the sides measure 3, 4 and 5 metres (or any multiple), the angle opposite the 5-metre side is exactly 90°. This Pythagorean triple (3-4-5) has been used since Ancient Egypt to mark out land and build pyramids.
GPS and 3D distances
The distance between two points in three-dimensional space is calculated using the theorem's extension: d = √(Δx² + Δy² + Δz²). GPS systems calculate your phone's position by triangulating distances to four satellites using precisely this formula, extended to include relativistic correction.
Screens and resolutions
Screen diagonals (TVs, laptops, phones) are measured in inches using the theorem: if a screen measures 16 × 9 inches, the diagonal is √(16² + 9²) = √(256 + 81) = √337 ≈ 18.4 inches. Manufacturers use the diagonal because it is the largest and most impressive number.
Results are exact for right-angled triangles. Make sure the 90° angle is correctly identified.
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