The Pythagorean Theorem in Everyday Life

The Pythagorean theorem says that in a right-angled triangle, the two shorter sides squared add up to the longest side squared: a² + b² = c². It is one of the most useful facts in school math, because right angles are everywhere: walls meet floors, screens are rectangles, and streets cross at corners.

Leaning a ladder

You need to reach a window 4 m up, and the ladder's foot must stand 1.5 m from the wall for safety. The ladder is the hypotenuse: √(4² + 1.5²) = √18.25 ≈ 4.27 m. A 4.5 m ladder will do the job.

TV and phone screen sizes

Screens are sold by their diagonal. A laptop screen 34.5 cm wide and 19.4 cm tall has a diagonal of √(1190.25 + 376.36) ≈ 39.6 cm, which is about 15.6 inches, a common laptop size.

Squaring a corner

Builders check right angles with the 3-4-5 rule. Measure 3 m along one wall and 4 m along the other. If the distance between those two marks is exactly 5 m, the corner is a true right angle, because 3² + 4² = 5².

Cutting across a park

A rectangular park is 120 m by 50 m. Walking around two sides is 170 m. Cutting straight across is √(14,400 + 2,500) = √16,900 = 130 m, saving 40 m.

Wheelchair ramps

A ramp must rise 0.5 m over a horizontal run of 6 m. The sloping surface is √(36 + 0.25) ≈ 6.02 m long, which tells you how much decking to buy.

Finding a shorter side

The formula also works backwards. If a 13 m cable runs from the top of a pole to a point 5 m from its base, the pole is √(169 − 25) = √144 = 12 m tall.

Tip The theorem only works with a right angle. For other triangles, use the law of cosines.

△Try it yourself: Pythagorean Theorem CalculatorMissing side of a right triangle from the other two, using a² + b² = c²Open →

Common questions