The Pythagorean Theorem with Real-World Examples

Understand a² + b² = c², find the hypotenuse or a missing side, and use it for ladders, screen sizes, squaring a corner and other real-world problems.

By the CalClu teamNovember 8, 20262 min read

The Pythagorean theorem links the three sides of any right-angled triangle. It's more than 2,000 years old and still one of the most useful formulas for everyday measuring.

The formula

a² + b² = c²

Here a and b are the two shorter sides that meet at the right angle, and c is the hypotenuse: the longest side, opposite the right angle.

c = √(a² + b²)

a = √(c² − b²)

Example 1: How long a ladder?

You need to reach a window 4 m up, with the ladder's foot 1 m from the wall. c = √(4² + 1²) = √17 ≈ 4.12 m, so a 4 m ladder won't reach.

Example 2: Screen sizes

TV and monitor sizes are diagonal measurements. A 16:9 screen 48 inches wide and 27 inches tall has a diagonal of √(48² + 27²) = √(2,304 + 729) = √3,033 ≈ 55.1 inches, sold as a 55-inch screen.

Example 3: Checking a corner is square

Builders use the 3-4-5 rule. Measure 3 units along one side and 4 along the other; if the diagonal between those two points is exactly 5, the corner is a true right angle. Scale it up for bigger layouts: 6-8-10 or 9-12-15.

Pythagorean triples

Whole-number solutions are handy to remember: 3-4-5, 5-12-13, 8-15-17 and 7-24-25. Any multiple of a triple works too.

Circle area: A = π × r². A 30 cm pizza (radius 15 cm) has about 707 cm², more than two 20 cm pizzas together (about 628 cm²).
Cylinder volume: V = π × r² × h. A can 6.6 cm across and 12 cm tall holds about π × 3.3² × 12 ≈ 410 cm³.

Try the calculators

Pythagorean Theorem (Hypotenuse) CalculatorLongest side of a right triangle from the two shorter sides a and b Circle Area CalculatorArea of a circle from its radius (half the diameter)Cylinder Volume CalculatorVolume of a cylinder, tank, can or pipe from its radius and height Square Root & PowersSquare roots, cube roots, nth roots and powers.

More tools in this area: Geometry calculators

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