What is the Pythagorean theorem?

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (c) equals the sum of the squares of the other two sides: a squared plus b squared equals c squared. This fundamental relationship only applies to right triangles (those with a 90-degree angle).

How do you find the hypotenuse?

To find the hypotenuse (c), use c = sqrt(a squared + b squared). For example, if side a = 3 and side b = 4, then c = sqrt(9 + 16) = sqrt(25) = 5. The hypotenuse is always the longest side, opposite the right angle.

How do you find a missing leg of a right triangle?

If you know the hypotenuse (c) and one leg (a), the other leg is b = sqrt(c squared minus a squared). For example, if c = 13 and a = 5, then b = sqrt(169 - 25) = sqrt(144) = 12.

What are Pythagorean triples?

Pythagorean triples are sets of three positive integers (a, b, c) that satisfy a squared + b squared = c squared. Common examples include (3, 4, 5), (5, 12, 13), (8, 15, 17), and (7, 24, 25). Any multiple of a Pythagorean triple is also a valid triple.

Pythagorean Theorem Calculator — Find Missing Sides

Pythagorean Theorem Calculator

Enter Any Two Sides

Leave one field empty to solve for the missing side. Side c is the hypotenuse (longest side).

Side a

Side b

Side c (hypotenuse)

Results

Side a

Side b

Side c (hypotenuse)

Area

Perimeter

Angle A (opposite a)

36.87°

Angle B (opposite b)

53.13°

Angle C (right angle)

90°

Triangle Diagram

Understanding the Pythagorean Theorem

The Pythagorean theorem is one of the most fundamental relationships in mathematics. It states that in any right triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides: a² + b² = c². This theorem, attributed to the ancient Greek mathematician Pythagoras, has been proven in hundreds of different ways and remains essential to geometry, trigonometry, and applied mathematics.

To find the hypotenuse when you know both legs, use c = √(a² + b²). To find a missing leg, rearrange to a = √(c² - b²). The classic 3-4-5 right triangle is the most well-known Pythagorean triple, but there are infinitely many integer solutions including 5-12-13, 8-15-17, and 7-24-25. Any multiple of a Pythagorean triple also satisfies the theorem, so 6-8-10 and 9-12-15 are valid as well.

Beyond pure mathematics, the Pythagorean theorem is used extensively in construction, navigation, physics, and computer graphics. Builders use the 3-4-5 rule to verify right angles. GPS systems calculate distances using extensions of this formula. Game developers compute distances between objects on screen. This calculator not only finds the missing side but also computes the triangle's area (A = ½ × a × b), perimeter, and both acute angles using inverse trigonometric functions, giving you a complete picture of the right triangle.

Formula

a² + b² = c²

Where:

a, b = the two legs of the right triangle
c = the hypotenuse (longest side, opposite the right angle)
To find hypotenuse: c = √(a² + b²)
To find a leg: a = √(c² − b²)

Example Calculation

Scenario: A right triangle has legs of 6 and 8. Find the hypotenuse.

Step 1: c² = 6² + 8² = 36 + 64 = 100
Step 2: c = √100 = 10
Result: The hypotenuse is 10 (this is a 3-4-5 triple scaled by 2)