Pythagorean Theorem Calculator

Calculate the missing side of a right triangle using the Pythagorean theorem: a² + b² = c². Enter any two sides to find the third side instantly.

Find Hypotenuse (c)

Enter the lengths of both legs (a and b) to calculate the hypotenuse (c).

a b c
c = √(a² + b²)
5.00 cm

Hypotenuse length with sides a = 3 cm and b = 4 cm

Step-by-step solution:
1. Apply Pythagorean theorem: c² = a² + b²
2. Substitute values: c² = 3² + 4² = 9 + 16 = 25
3. Take square root: c = √25 = 5.00 cm

Find Side a (leg)

Enter the hypotenuse (c) and the other leg (b) to calculate side a.

a b c
a = √(c² - b²)
3.00 cm

Side a length with hypotenuse c = 5 cm and side b = 4 cm

Step-by-step solution:
1. Rearrange Pythagorean theorem: a² = c² - b²
2. Substitute values: a² = 5² - 4² = 25 - 16 = 9
3. Take square root: a = √9 = 3.00 cm

Find Side b (leg)

Enter the hypotenuse (c) and the other leg (a) to calculate side b.

a b c
b = √(c² - a²)
4.00 cm

Side b length with hypotenuse c = 5 cm and side a = 3 cm

Step-by-step solution:
1. Rearrange Pythagorean theorem: b² = c² - a²
2. Substitute values: b² = 5² - 3² = 25 - 9 = 16
3. Take square root: b = √16 = 4.00 cm

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Pythagorean Theorem Calculator: Complete Guide to Finding Triangle Sides

The Pythagorean theorem calculator is an essential tool for finding the missing side of a right triangle. Named after the ancient Greek mathematician Pythagoras, this fundamental theorem states that in a right triangle, the square of the hypotenuse equals the sum of squares of the other two sides: a² + b² = c².

What is the Pythagorean Theorem?

The Pythagorean theorem is one of the most important principles in geometry. It establishes a relationship between the three sides of a right triangle, where:

How to Use the Pythagorean Theorem Calculator

Finding the Hypotenuse

When you know both legs (a and b), use the formula c = √(a² + b²) to find the hypotenuse. This is the most common application of the Pythagorean theorem.

Finding a Leg

When you know the hypotenuse and one leg, rearrange the formula to find the missing leg:

Common Applications of the Pythagorean Theorem

Construction and Architecture

Builders use the Pythagorean theorem to ensure structures are square and to calculate diagonal distances. The famous 3-4-5 triangle is commonly used to check right angles in construction.

Navigation and GPS

GPS systems use the Pythagorean theorem to calculate direct distances between two points, helping determine the shortest route.

Sports and Recreation

The theorem helps calculate distances in sports fields, determine diagonal measurements for courts, and optimize playing strategies.

Engineering and Design

Engineers apply the Pythagorean theorem in designing structures, calculating forces, and determining optimal angles for various applications.

Step-by-Step Pythagorean Theorem Examples

Example 1: Finding the Hypotenuse

Given: a = 6 units, b = 8 units

  1. Apply the formula: c² = a² + b²
  2. Substitute: c² = 6² + 8² = 36 + 64 = 100
  3. Take the square root: c = √100 = 10 units

Example 2: Finding a Leg

Given: c = 13 units, b = 5 units

  1. Rearrange the formula: a² = c² - b²
  2. Substitute: a² = 13² - 5² = 169 - 25 = 144
  3. Take the square root: a = √144 = 12 units

Famous Pythagorean Triples

Pythagorean triples are sets of three positive integers that satisfy the Pythagorean theorem. Common examples include:

Tips for Using the Pythagorean Theorem

Verify Right Triangles

The Pythagorean theorem only applies to right triangles. Always confirm you're working with a 90-degree angle before applying the formula.

Unit Consistency

Ensure all measurements use the same units. Our calculator supports conversions between centimeters, meters, inches, and feet.

Check Your Work

After calculating, verify your answer by substituting back into the original equation to ensure a² + b² = c².

Common Mistakes to Avoid

Identifying the Hypotenuse

The hypotenuse is always the longest side and is always opposite the right angle. Don't confuse it with the legs.

Calculation Errors

Be careful with squaring and square root operations. Double-check your arithmetic, especially with decimals.

Unit Confusion

Always express your final answer in the appropriate units. If calculating area, the result should be in square units.

Advanced Applications

3D Distance Formula

The Pythagorean theorem extends to three dimensions: d = √(x² + y² + z²), useful in 3D modeling and physics.

Coordinate Geometry

Find distances between points on a coordinate plane using the distance formula, which is based on the Pythagorean theorem.

Trigonometry Foundation

The Pythagorean theorem forms the basis for trigonometric ratios (sine, cosine, tangent) and the Pythagorean identity: sin²θ + cos²θ = 1.

Conclusion

The Pythagorean theorem calculator is an invaluable tool for students, professionals, and anyone working with right triangles. Whether you're solving homework problems, designing buildings, or calculating distances, understanding and applying this fundamental theorem opens doors to countless mathematical and practical applications. Master the formula a² + b² = c², and you'll have a powerful tool for solving geometric problems with confidence.