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Pythagorean Theorem

Solve for side A, B, or hypotenuse C.

Triangle Inputs

Select which side to solve for and input the other two.

Result —

Calculation Breakdown:

Enter valid values to compute the missing side.

About this topic

The Pythagorean Theorem In any right triangle, a² + b² = c².

The theorem states that the square of the hypotenuse equals the sum of the squares of the two legs. It is the foundation of Euclidean geometry and trigonometry.

Key formulas & rules

  • •Formula: a² + b² = c²
  • •c is the hypotenuse (opposite the right angle)
  • •a and b are the two legs
  • •Works only for right triangles (90° angle)
  • •Converse is also true: if a² + b² = c², it's a right triangle
  • •Generalizes to the Law of Cosines

Examples

Input Output Note
a=3, b=4 c=5 Classic 3-4-5 triangle
a=5, b=12 c=13 Common triple
a=8, b=15 c=17 Another triple
a=1, b=1 c=√2 Isosceles right

Real-world uses

Construction Navigation Screen resolution
Pythagorean Triples Integer solutions to a² + b² = c².

A Pythagorean triple is a set of three positive integers that satisfy the theorem. They are useful for quick mental math and appear throughout number theory.

Key formulas & rules

  • •Primitive triples have gcd(a,b,c) = 1
  • •Euclid's formula generates all primitive triples
  • •(3, 4, 5) is the smallest triple
  • •Every multiple of a triple is also a triple
  • •Common triples: (5,12,13), (8,15,17), (7,24,25)
  • •Infinite number of primitive triples exist

Examples

Input Output Note
(3, 4, 5) 3² + 4² = 5² Primitive
(6, 8, 10) 2 × (3, 4, 5) Multiple
(5, 12, 13) 25 + 144 = 169 Primitive
(20, 21, 29) 400 + 441 = 841 Primitive

Real-world uses

Carpentry Quick math Number theory
Visual Proof How a² + b² literally equals c².

The theorem has over 370 known proofs. The most famous arranges four identical right triangles inside a square to visually demonstrate that the areas of the two smaller squares equal the area of the larger one.

Key formulas & rules

  • •Rearrangement proof uses 4 triangles
  • •Algebraic proof: (a+b)² = c² + 2ab
  • •Similar triangles proof (Euclid)
  • •Over 370 distinct published proofs
  • •Pythagoras credited, but not the originator
  • •Babylonians knew it 1000 years earlier

Examples

Input Output Note
Area method (a+b)² = c² + 2ab Algebra proof
Rearrange 4 triangles Visual identity Geometric proof
Similar triangles a² = c·x Euclid's proof
Garfield's trapezoid ½(a+b)² Presidential proof

Real-world uses

Mathematics education Theorem proving Geometry history

Modes & formulas

1

Solve for hypotenuse (c)

• Finds the longest side of a right triangle. • Enter the lengths of the two shorter sides (a and b). • The calculator squares both sides, adds them, and takes the square root.
c = √(a² + b²)

This works only if the angle opposite the hypotenuse is exactly 90 degrees.

2

Solve for a leg (a or b)

• Finds the length of one of the shorter sides. • Enter the hypotenuse and the other leg. • The calculator squares both, subtracts the leg from the hypotenuse, and takes the square root.
a = √(c² - b²)

The hypotenuse must always be the longest side, so the tool will throw an error if c is less than b.

How to use it

Select which side you want to find. Enter the two known sides. The tool provides the exact length and a dynamic visual graph.

Frequently asked questions

What is the hypotenuse?
The longest side of a right triangle, located opposite the 90-degree angle.
Does this theorem work for non-right triangles?
No. The Pythagorean Theorem only applies to right-angled triangles. For other triangles, use the Law of Cosines.
What is the Pythagorean Theorem?
A fundamental principle in geometry stating that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides (a² + b² = c²).
What are the legs of a right triangle?
The two shorter sides that meet at the 90-degree angle.
What is a Pythagorean triple?
A set of three positive integers that perfectly satisfy the theorem, such as 3, 4, and 5 (since 3² + 4² = 5²).
Can a right triangle have equal legs?
Yes, it is called an isosceles right triangle (45-45-90 triangle). The hypotenuse will be leg × √2.

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Read the full Pythagorean Theorem guide — examples & 18 FAQs