Pythagorean Theorem Calculator - Find Any Side of a Right Triangle

Solve for any missing side of a right triangle using the Pythagorean theorem with our free online calculator. Enter two sides and instantly find the third with step-by-step working.

Side a

Side b

Examples:

Understanding the Pythagorean Theorem

The Pythagorean theorem is one of the most famous and useful relationships in mathematics. 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.

This 2,500-year-old theorem isn't just abstract math - it's used in construction, navigation, computer graphics, physics, and countless real-world applications. Every time you calculate a diagonal distance or check if a corner is square, you're using Pythagoras' discovery.

The Pythagorean Theorem Formula

a² + b² = c²

Where:

• a and b are the lengths of the legs (the sides forming the right angle)

• c is the length of the hypotenuse (the side opposite the right angle)

• The hypotenuse is always the longest side

Finding the Hypotenuse

c = √(a² + b²)

When you know both legs, square them, add, then take the square root.

Finding a Leg

a = √(c² - b²)

When you know the hypotenuse and one leg, subtract squares, then take the square root.

Worked Examples

Example 1: The Classic 3-4-5 Triangle

Find the hypotenuse when the legs are 3 and 4.

a = 3, b = 4

c² = 3² + 4²

c² = 9 + 16 = 25

c = √25 = 5

The hypotenuse is 5 units

This is the famous 3-4-5 right triangle used by carpenters!

Example 2: Finding a Missing Leg

The hypotenuse is 13 and one leg is 5. Find the other leg.

c = 13, a = 5

b² = c² - a²

b² = 13² - 5² = 169 - 25 = 144

b = √144 = 12

The missing leg is 12 units

Another Pythagorean triple: 5-12-13

Example 3: Real-World Application - Ladder Problem

A 15-foot ladder leans against a wall. The base is 9 feet from the wall. How high up the wall does the ladder reach?

Hypotenuse (ladder) = 15 ft

Base distance = 9 ft

Height² = 15² - 9² = 225 - 81 = 144

Height = √144 = 12 ft

The ladder reaches 12 feet up the wall

Example 4: Diagonal Distance

A rectangular field is 100m by 75m. What's the diagonal distance across the field?

a = 100m, b = 75m

c² = 100² + 75² = 10,000 + 5,625 = 15,625

c = √15,625 = 125m

The diagonal is 125 meters

Walking diagonally saves 50m compared to walking the edges!

Example 5: Non-Integer Result

Find the hypotenuse when legs are 1 and 1.

a = 1, b = 1

c² = 1² + 1² = 1 + 1 = 2

c = √2 ≈ 1.4142

The hypotenuse is √2 (approximately 1.4142)

This is an irrational number - the decimal never ends or repeats!

Quick Fact

Although named after Greek mathematician Pythagoras (570-495 BCE), the theorem was known to Babylonian mathematicians over 1,000 years earlier. A clay tablet called Plimpton 322 (1800 BCE) contains Pythagorean triples. The theorem has over 370 different proofs, including one by U.S. President James Garfield in 1876 - while he was still a Congressman!

Frequently Asked Questions

Does the Pythagorean theorem work for all triangles?

No, only for right triangles (triangles with a 90° angle). For other triangles, you need the Law of Cosines, which is a generalization of the Pythagorean theorem. If the triangle isn't right, a² + b² ≠ c².

What are Pythagorean triples?

Sets of three whole numbers that satisfy a² + b² = c². Common examples: 3-4-5, 5-12-13, 8-15-17, 7-24-25. Any multiple also works (6-8-10, 9-12-15). These are useful for quick mental calculations.

How do I know which side is the hypotenuse?

The hypotenuse is always opposite the right angle and is always the longest side. In diagrams, it's often labeled 'c'. If you're not sure which angle is 90°, the hypotenuse is the side that doesn't touch the right angle.

Can the result be a decimal or irrational number?

Yes! Most right triangles don't have whole number sides. For example, if both legs are 1, the hypotenuse is √2, which is irrational (approximately 1.41421356...). The decimal goes on forever without repeating.

What's the converse of the Pythagorean theorem?

If a² + b² = c² for a triangle's sides, then the triangle MUST be a right triangle. This lets you test if a triangle is right-angled without measuring angles. Useful in construction for checking square corners.

Where is the Pythagorean theorem used in real life?

Construction (checking square corners, calculating roof slopes), navigation (shortest distance), surveying, computer graphics (distance between pixels), physics (vector components), GPS triangulation, and anywhere diagonal distances matter.

Other Free Tools

Search tools