Pythagorean Theorem Calculator

The Pythagorean theorem is the most famous formula in geometry. It applies to any right triangle (a triangle with exactly one 90° angle) and gives the relationship between the three side lengths.

The formula: a² + b² = c²

Where a and b are the two legs (the sides meeting at the right angle) and c is the hypotenuse (the side opposite the right angle, which is always the longest).

Rearranged forms:

• Find the hypotenuse: c = √(a² + b²)
• Find a leg: a = √(c² − b²) or b = √(c² − a²)

The ancient trick that still works on construction sites: A rope or tape measure cut into a 3-4-5 ratio forms a perfect right angle. Stretch 3 units along one wall, 4 units along the other, and if the diagonal measures exactly 5 units, the corner is square. Carpenters and surveyors have been using this since before formal geometry existed.

Pythagorean triples are sets of three positive integers (a, b, c) that satisfy a² + b² = c². The classic ones:

• 3, 4, 5 → 9 + 16 = 25 ✓
• 5, 12, 13 → 25 + 144 = 169 ✓
• 8, 15, 17 → 64 + 225 = 289 ✓
• 7, 24, 25 → 49 + 576 = 625 ✓
• 9, 40, 41 → 81 + 1,600 = 1,681 ✓
• 20, 21, 29
• 12, 35, 37

Any whole-number multiple of a triple is also a triple. 6-8-10, 9-12-15, and 30-40-50 all derive from 3-4-5.

The converse is just as useful: if a triangle’s sides satisfy a² + b² = c², then the angle opposite c must be exactly 90°. Construction inspectors use this to verify framing. Out-of-square corners cause cabinets to bind, doors to stick, and floors to slope.

Worked example, a ladder problem: A 13 ft ladder rests against a wall with its base 5 ft from the wall. How high up does the ladder reach?

The ladder is the hypotenuse (c = 13). The base distance is one leg (b = 5). The height is the other leg (a):

a² = c² − b² = 169 − 25 = 144 a = 12 ft

Worked example, diagonal distance: You want to know the straight-line distance between two points on a coordinate plane: (1, 2) and (4, 6).

The horizontal separation is 3 (= 4 − 1) and the vertical separation is 4 (= 6 − 2). These form the legs of a right triangle whose hypotenuse is the diagonal you want:

d = √(3² + 4²) = √25 = 5 units

This is exactly the distance formula. It is the Pythagorean theorem dressed up for coordinate geometry.

In 3D: The theorem extends naturally. The diagonal of a rectangular box with sides a, b, c is √(a² + b² + c²). This shows up in physics (vector magnitudes), graphics (camera distance), navigation (3D position differences), and structural engineering (diagonal bracing).

When it does not apply: The Pythagorean theorem only works for right triangles. For any other triangle, use the Law of Cosines: c² = a² + b² − 2ab·cos(C), which reduces to a² + b² = c² when C = 90° (cos 90° = 0).