Pythagoras' theorem is one of the most useful formulas in practical geometry — from building a square corner to calculating screen sizes. Here's the formula, proofs, and real-world examples.
Pythagoras' theorem connects the three sides of any right-angled triangle. It's been used for 4,000 years of construction, navigation, and engineering — and it comes up more often in everyday life than most people realise.
Finding the hypotenuse: A right triangle has legs of 6cm and 8cm. Find the hypotenuse.
Finding a shorter side: A ladder 5m long leans against a wall with its foot 3m from the base. How high up the wall does it reach?
Some right triangles have all three sides as whole numbers — these are called Pythagorean triples and are useful for builders checking square corners without calculation:
The 3:4:5 triple is the most commonly used in construction for squaring corners — measure 3 units along one wall, 4 units along the adjacent wall, and the diagonal should be exactly 5 units if the corner is perfectly square.
Screen sizes: TV and monitor sizes are quoted as the diagonal measurement. If a TV is advertised as 55 inches (diagonal) with a 16:9 aspect ratio, you can calculate the actual width and height:
Stair calculation: If a staircase rises 2.4m and runs 3.2m horizontally, the total stringer length is:
GPS distance: For short distances, GPS coordinates can be approximated as a right triangle. If you're 3km east and 4km north of a destination, the straight-line distance is: