Pythagorean Theorem Calculator

Find the missing side of a right triangle. Calculate hypotenuse or legs using a^2 + b^2 = c^2.

Which side to calculate

One side of the right angle

Other side of the right angle

Longest side (opposite right angle)

Known leg

Longest side

Known leg

Pythagorean Theorem: a^2 + b^2 = c^2\n\nTo find hypotenuse: c = sqrt(a^2 + b^2)\nTo find leg a: a = sqrt(c^2 - b^2)\nTo find leg b: b = sqrt(c^2 - a^2)\n\nWhere:\na, b = legs (sides forming right angle)\nc = hypotenuse (longest side)
Example 1 (Find hypotenuse):\na = 3, b = 4\nc = sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt25 = 5\nTriple: 3-4-5 ✓\n\nExample 2 (Find leg):\nc = 13, b = 12\na = sqrt(13^2 - 12^2) = sqrt(169 - 144) = sqrt25 = 5\nTriple: 5-12-13 ✓\n\nExample 3:\na = 5, b = 5\nc = sqrt(25 + 25) = sqrt50 = 7.071

What is the Pythagorean theorem?

For any right triangle: a^2 + b^2 = c^2, where a and b are the legs (sides forming the right angle) and c is the hypotenuse (longest side opposite right angle). Example: If legs are 3 and 4, then 3^2 + 4^2 = 9 + 16 = 25, so c = sqrt25 = 5.

How do I find a missing side of a right triangle?

To find hypotenuse c: c = sqrt(a^2 + b^2). To find leg a: a = sqrt(c^2 - b^2). To find leg b: b = sqrt(c^2 - a^2). Example: If c=10 and a=6, then b = sqrt(100-36) = sqrt64 = 8.

What are Pythagorean triples?

Integer sets that satisfy a^2 + b^2 = c^2. Common triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25. Multiples also work: 6-8-10, 9-12-15, etc. Useful for quick calculations without decimals. Any multiple of a triple is also a triple.

Does the Pythagorean theorem work for all triangles?

NO - only for RIGHT triangles (one 90deg angle). For other triangles, use Law of Cosines: c^2 = a^2 + b^2 - 2ab*cos(C). If a^2 + b^2 > c^2, triangle is acute. If a^2 + b^2 < c^2, triangle is obtuse. If a^2 + b^2 = c^2, triangle is right.

What are real-world uses of the Pythagorean theorem?

Construction: squaring corners, roof pitch, stairs. Navigation: distance calculations, GPS. Engineering: structural supports, bracing. Sports: baseball diamond distances. Screen sizes (diagonal). Ladder safety (base distance from wall). Any scenario with right angles.