Pythagorean Theorem Calculator

Solve right triangle problems with precision and clarity 📐

c² = a² + b²

Finding hypotenuse (c) from legs a and b

Leg a:

Leg b:

a² = c² – b²

Finding leg (a) from hypotenuse c and leg b

Hypotenuse c:

Known leg b:

Does a² + b² = c²?

Verify if three sides form a right triangle

Side 1:

Side 2:

Side 3:

The Pythagorean theorem stands as one of mathematics most elegant and practical discoveries, bridging ancient geometry with modern engineering. From architects designing stable structures to GPS satellites calculating distances, this 2,500-year-old principle remains indispensable in contemporary problem-solving. Whether you’re a construction professional ensuring square corners, a game developer calculating collision detection, or a student exploring geometric relationships, mastering Pythagorean calculations opens doors to spatial reasoning and mathematical confidence.

📐 Geometric Connections

The Pythagorean theorem connects to broader mathematical concepts. Expand your geometric toolkit with our Triangle Calculator for general triangle properties and Distance Calculator for coordinate plane applications.

a² + b² = c²

Table of Contents

Understanding the Pythagorean Theorem

This fundamental geometric principle establishes the relationship between the sides of a right triangle, where the square of the hypotenuse equals the sum of squares of the other two sides.

Theorem Components

Legs (a, b): The two sides that form the right angle
Hypotenuse (c): The longest side, opposite the right angle
Right Angle: The 90-degree angle where the legs meet

Practical Applications

Construction & Architecture: Ensuring square foundations, calculating diagonal bracing, determining roof pitch angles

Navigation & GPS: Computing straight-line distances, triangulating positions, optimizing travel routes

Engineering: Structural analysis, force vector calculations, mechanical design optimization

Computer Graphics: Distance calculations, collision detection, 3D rendering computations

Solution Methods

Finding Hypotenuse: c = √(a² + b²)

Finding Leg: a = √(c² – b²)

Verification: Check if a² + b² = c² (within tolerance)

Frequently Asked Questions

Why does the Pythagorean theorem only work for right triangles?

The theorem is specifically derived from the geometric properties of right triangles. The relationship a² + b² = c² emerges from the way squares can be arranged around a right triangle. For other triangle types, use the Law of Cosines, which generalizes this relationship.

How accurate should my Pythagorean calculations be?

Accuracy depends on your application. For construction, measurements to the nearest 1/16 inch are often sufficient. For engineering calculations, you might need several decimal places. Always consider the precision of your input measurements when determining output accuracy.

What are Pythagorean triples?

Pythagorean triples are sets of three positive integers that satisfy the theorem. The most famous is (3, 4, 5), but there are infinite triples like (5, 12, 13) and (8, 15, 17). These are particularly useful in construction for creating exact right angles without measuring tools.

Can I use the theorem in three dimensions?

Yes! The theorem extends to 3D as: d² = x² + y² + z², where d is the spatial diagonal and x, y, z are the three perpendicular dimensions. This is fundamental for 3D graphics, architecture, and engineering applications.

How do I know if my triangle is actually a right triangle?

Measure all three sides and test if a² + b² = c² (where c is the longest side). Due to measurement precision, look for results very close to equality rather than exact matches. A difference of less than 1% usually indicates a right triangle within measurement tolerance.