Pythagorean Theorem Calculator — Free a² + b² = c² Solver | AllInOneTools
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Pythagorean Theorem Calculator

Enter any two sides of a right triangle and find the missing side. See a visual diagram, angles, area, perimeter, and step-by-step solution.

Side a (leg)
Side b (leg)
Side c (hypotenuse)
Enter any 2 sides. Leave the unknown side blank.
a² + b² = c²
📑 Step-by-Step Solution

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The Pythagorean Theorem: Foundation of Geometry

The Pythagorean theorem is arguably the most famous equation in all of mathematics: a² + b² = c². It states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides. Attributed to the Greek mathematician Pythagoras (c. 570-495 BC), though evidence suggests Babylonian mathematicians knew the relationship over a thousand years earlier, this theorem is fundamental to geometry, trigonometry, physics, engineering, architecture, and navigation.

The Formula and Its Variations

Standard Form:
  a² + b² = c²

Solving for each side:
  c = √(a² + b²)   (find hypotenuse)
  a = √(c² - b²)   (find leg a)
  b = √(c² - a²)   (find leg b)

Triangle Properties:
  Area = (a × b) / 2
  Perimeter = a + b + c
  Angle A = arctan(a/b) in degrees
  Angle B = arctan(b/a) in degrees
  Altitude h = (a × b) / c

Common Pythagorean Triples:
  3, 4, 5 | 5, 12, 13 | 8, 15, 17
  7, 24, 25 | 20, 21, 29 | 9, 40, 41

Real-World Applications

The Pythagorean theorem is used constantly in construction and architecture to ensure walls are perpendicular (the 3-4-5 method). Navigation uses it to calculate straight-line distances between two points. In computer graphics, pixel distances are calculated using the theorem's 2D extension (and its 3D version: d = √(x² + y² + z²)). Surveying, astronomy, GPS technology, and physics (calculating resultant vectors) all rely on this fundamental relationship.

The 3-4-5 Construction Method
Builders use Pythagorean triples to check right angles without special tools. Measure 3 units along one wall and 4 units along the perpendicular wall. If the diagonal distance is exactly 5 units, the corner is a perfect 90-degree angle. Multiples also work: 6-8-10, 9-12-15, 12-16-20, etc. This technique has been used for thousands of years in construction worldwide.

Frequently Asked Questions

What is the Pythagorean theorem? ▾
In a right triangle, the square of the hypotenuse equals the sum of squares of the two legs: a² + b² = c². It only applies to right triangles (one 90-degree angle).
How do I find the hypotenuse? ▾
c = √(a² + b²). Square both legs, add them, take the square root. Example: sides 3 and 4 give c = √(9+16) = √25 = 5.
How do I find a missing leg? ▾
a = √(c² - b²). Square the hypotenuse, subtract the square of the known leg, take the square root. Example: c=13, b=12 gives a = √(169-144) = √25 = 5.
What are Pythagorean triples? ▾
Sets of three positive integers that satisfy a² + b² = c². Common examples: (3,4,5), (5,12,13), (8,15,17), (7,24,25). Any multiple of a triple is also a triple.
Does it work for non-right triangles? ▾
No. For non-right triangles, use the Law of Cosines: c² = a² + b² - 2ab·cos(C). The Pythagorean theorem is the special case where angle C = 90° (cos 90° = 0).
What is the distance formula? ▾
d = √((x2-x1)² + (y2-y1)²). It is a direct application of the Pythagorean theorem on the coordinate plane, treating horizontal and vertical distances as legs.