Free Pythagorean Theorem Calculator

Calculate the missing side of a right triangle instantly using the Pythagorean theorem. Enter any two sides to find the third, or verify if three sides form a valid right triangle.

What is the Pythagorean Theorem?

The Pythagorean theorem is one of the most fundamental principles in geometry. Named after the ancient Greek mathematician Pythagoras, it describes the relationship between the three sides of a right triangle. A right triangle is any triangle that contains a 90-degree angle (a right angle).

The theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides (called legs). This simple yet powerful relationship forms the basis for countless calculations in mathematics, physics, engineering, and everyday life.

The Formula a² + b² = c² Explained

The Pythagorean theorem is expressed by the formula:

a² + b² = c²

Where:

• a and b are the lengths of the two legs (the sides that form the right angle)
• c is the length of the hypotenuse (the longest side, opposite the right angle)

This formula can be rearranged to solve for any missing side:

• To find the hypotenuse: c = √(a² + b²)
• To find a leg: a = √(c² - b²) or b = √(c² - a²)

The key insight is that the area of a square built on the hypotenuse equals the combined area of squares built on the other two sides.

How to Use This Calculator

Our Pythagorean theorem calculator supports three modes of operation:

Find the Hypotenuse (c)

1. Enter the values of sides a and b (the two legs)
2. The calculator computes c = √(a² + b²)
3. View the step-by-step solution and result

Find a Missing Leg (a or b)

1. Enter one known leg and the hypotenuse c
2. The calculator computes the missing leg using b = √(c² - a²)
3. The leg value must be less than the hypotenuse

Verify a Right Triangle

1. Enter all three side lengths
2. The calculator checks whether a² + b² = c² holds true
3. See whether the sides form a valid right triangle with full working

Common Pythagorean Triples

A Pythagorean triple is a set of three positive integers that satisfy the equation a² + b² = c². These are incredibly useful in mathematics and appear frequently in problems and real-world applications.

Any multiple of a Pythagorean triple is also a Pythagorean triple. For example, multiplying 3-4-5 by 2 gives 6-8-10, and by 3 gives 9-12-15. Our calculator automatically detects which Pythagorean triple your triangle belongs to.

Proof of the Pythagorean Theorem

There are hundreds of known proofs of the Pythagorean theorem. Here is one of the most intuitive:

The Area Proof: Construct a large square with side length (a + b). Inside this square, arrange four identical right triangles, each with legs a and b, so that they form a smaller inner square with side length c.

The area of the large square equals (a + b)². This area can also be expressed as the sum of the four triangles plus the inner square: 4 × (½ab) + c².

Setting these equal:

(a + b)² = 4 × (½ab) + c²

a² + 2ab + b² = 2ab + c²

a² + b² = c²

This elegant proof demonstrates why the theorem holds for every right triangle without exception.

Real-World Applications of the Pythagorean Theorem

The Pythagorean theorem is not just an abstract mathematical concept. It has practical applications across many fields:

Construction and Carpentry

Builders use the 3-4-5 rule to ensure corners are perfectly square. By measuring 3 feet along one wall and 4 feet along the adjacent wall, the diagonal should measure exactly 5 feet if the corner is a true right angle. This technique has been used for thousands of years.

Navigation and Distance

Pilots and sailors use the theorem to calculate the shortest distance between two points. If you know how far east and how far north you need to travel, the Pythagorean theorem gives you the straight-line distance.

Architecture

Architects rely on the theorem to calculate diagonal lengths, determine roof pitches, and ensure structural integrity. When designing buildings with right-angle components, the theorem provides essential measurements.

Surveying

Land surveyors use the theorem to calculate distances that cannot be measured directly. By establishing right triangles with known measurements, they can determine the positions of boundaries and features.

Physics

In physics, the theorem is used to resolve vector components, calculate resultant forces, and determine displacement. Velocity components in two dimensions are often combined using the Pythagorean theorem.

Computer Graphics

Game developers and graphic designers use the theorem constantly to calculate distances between objects on screen. Collision detection, camera positioning, and movement calculations all rely on this fundamental relationship.

How to Find the Hypotenuse

To find the hypotenuse of a right triangle when you know both legs:

1. Square both leg values: calculate a² and b²
2. Add the squared values: a² + b²
3. Take the square root of the sum: c = √(a² + b²)

Example: Find the hypotenuse when a = 6 and b = 8

• a² = 36
• b² = 64
• a² + b² = 100
• c = √100 = 10

The hypotenuse is 10. This is a multiple of the 3-4-5 Pythagorean triple.

How to Find a Missing Leg

To find a missing leg when you know one leg and the hypotenuse:

1. Square the hypotenuse: calculate c²
2. Square the known leg: calculate a²
3. Subtract: c² - a²
4. Take the square root: b = √(c² - a²)

Example: Find the missing leg when a = 5 and c = 13

• c² = 169
• a² = 25
• c² - a² = 144
• b = √144 = 12

The missing leg is 12. This is the classic 5-12-13 Pythagorean triple.

Frequently Asked Questions

What is the Pythagorean theorem?

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. It is expressed as a² + b² = c², where c is the hypotenuse (longest side) and a and b are the legs.

How do I find the hypotenuse?

To find the hypotenuse, square both legs, add them together, and take the square root of the result. Use the formula c = √(a² + b²). For example, if a = 3 and b = 4, then c = √(9 + 16) = √25 = 5.

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

If you know the hypotenuse and one leg, subtract the square of the known leg from the square of the hypotenuse, then take the square root. Use a = √(c² - b²) or b = √(c² - a²).

What is a Pythagorean triple?

A Pythagorean triple is a set of three positive integers (a, b, c) that satisfy the equation a² + b² = c². The most well-known example is (3, 4, 5). Any integer multiple of a Pythagorean triple is also a valid triple.

Does the Pythagorean theorem work for all triangles?

No. The Pythagorean theorem only applies to right triangles (triangles with one 90-degree angle). For non-right triangles, you would use the Law of Cosines instead: c² = a² + b² - 2ab·cos(C).

How do I prove the Pythagorean theorem?

There are over 300 known proofs. The most common involve constructing squares on each side of the triangle and comparing areas, or using algebra with four copies of the triangle arranged inside a larger square. See the proof section above for a detailed explanation.

Can the hypotenuse equal one of the legs?

No. The hypotenuse is always the longest side of a right triangle. By definition, it must be longer than either leg. If you find that your hypotenuse is equal to or shorter than a leg, there is an error in the calculation.

What is the converse of the Pythagorean theorem?

The converse states that if the square of the longest side of a triangle equals the sum of the squares of the other two sides, then the triangle is a right triangle. This is used in the "Verify Triangle" mode of our calculator.

Why Use Our Pythagorean Theorem Calculator

• Free to use - No registration or sign-up required
• Three calculation modes - Find hypotenuse, find leg, or verify a triangle
• Step-by-step solutions - See every calculation step clearly
• Pythagorean triple detection - Automatically identifies known triples
• Visual right triangle - Interactive SVG diagram with labeled sides
• Quick examples - Load common triples with one click
• Mobile friendly - Works perfectly on any device
• Instant results - Real-time calculation as you type