How do I find the hypotenuse?

Use c = sqrt(a^2 + b^2). Example: a = 3, b = 4 gives c = 5.

How do I find a missing leg?

Use a = sqrt(c^2 - b^2) or b = sqrt(c^2 - a^2). The hypotenuse must be the longest side.

What are Pythagorean triples?

They are integer triples that satisfy a^2 + b^2 = c^2, such as (3, 4, 5) and (5, 12, 13).

What are real-world applications of the Pythagorean theorem?

It is used in construction, surveying, navigation, and computer graphics to compute straight-line distances or verify right angles.

Pythagorean Theorem Calculator - Solve Right Triangles

Q: What is the Pythagorean theorem?

In a right triangle, the square of the hypotenuse equals the sum of the squares of the legs: a^2 + b^2 = c^2.

Pythagorean Theorem Calculator

Solve right triangle problems using the Pythagorean theorem. Find missing sides with step-by-step solutions and learn the mathematical concepts behind this fundamental theorem.

100% FreeStep-by-step SolutionsVisual Triangle

Pythagorean Theorem Calculator

a^2 + b^2 = c^2

where a and b are legs, and c is the hypotenuse

Select Calculation Mode:

Side a (leg)

First leg of the triangle

Side b (leg)

Second leg of the triangle

Side c (hypotenuse)

Hypotenuse (longest side)

Instructions: Enter any two sides to calculate the third. Leave one field empty for the side you want to find.

Try These Examples

Click on any example to automatically fill the calculator

Example

Classic 3-4-5 triangle

a: 3

b: 4

c: ?

Example

5-12-13 Pythagorean triple

a: 5

b: ?

c: 13

Example

Find missing leg

a: ?

b: 8

c: 10

Example

45-45-90 triangle

a: 1

b: 1

c: ?

What is the Pythagorean Theorem?

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of squares of the other two sides.

a^2 + b^2 = c^2

Key Components:

a, b: The legs (sides forming the right angle)
c: The hypotenuse (longest side, opposite right angle)
Right Triangle: Must have a 90° angle

Historical Note: While named after Pythagoras (c. 570-495 BC), this theorem was known to ancient civilizations including Babylonians and Indians centuries earlier.

How to Apply the Theorem

Three Cases:

Finding the hypotenuse (c):

c = \sqrt{a^2 + b^2}

Finding a leg (a):

a = \sqrt{c^2 - b^2}

Finding a leg (b):

b = \sqrt{c^2 - a^2}

Common Pythagorean Triples:

3-4-5, 5-12-13, 8-15-17, 7-24-25

9-12-15, 12-16-20, 15-20-25

And their multiples...

Real-World Applications

Construction & Architecture

Calculating diagonal braces in framing
Determining rafter lengths for roofs
Ensuring square corners in foundations
Planning staircase dimensions

Engineering & Design

Navigation and GPS calculations
Computer graphics and game development
Robotics and movement planning
Structural engineering analysis

Everyday Life

Calculating TV screen sizes
Finding shortest distances on maps
Sports field measurements
Home improvement projects

Famous Proofs of the Pythagorean Theorem

Euclid's Proof (Geometric)

Uses the areas of squares constructed on each side of the triangle, showing that the area of the square on the hypotenuse equals the sum of areas of squares on the other two sides.

President Garfield's Proof

Uses a trapezoid formed by arranging two copies of the right triangle, comparing its area calculated in two different ways to derive the theorem.

Chinese Proof (Zhao Shuang)

Arranges four congruent right triangles around a central square, showing the relationship through area calculations.

Algebraic Proof

Uses coordinate geometry and the distance formula to prove the theorem algebraically, connecting it to analytic geometry.

How GPS Finds You in 1 Second: The 3D Pythagorean Theorem

Every time your phone shows your location, it's solving the Pythagorean theorem in three dimensions. GPS satellites don't measure your position directly — they measuredistances, and your phone calculates where those distances intersect.

From 2D to 3D: The Extended Formula

In two dimensions, the distance between points is:

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

In three dimensions (adding altitude), it becomes:

d = \sqrt{\Delta x^2 + \Delta y^2 + \Delta z^2}

This is just the Pythagorean theorem applied twice: first in the horizontal plane, then vertically.

🛰️ Why GPS Needs 4 Satellites

Each satellite gives you a sphere of possible positions (all points at distance d from the satellite). Two spheres intersect in a circle. Three spheres narrow it to two points. The fourth satellite resolves the ambiguity and corrects for clock errors — giving you accuracy within 3–5 meters.

🔢 Real Example

A drone is at coordinates (100, 200, 50) meters. The control station is at (400, 600, 0). What's the straight-line distance?

d = \sqrt{300^2 + 400^2 + 50^2}

= \sqrt{90000 + 160000 + 2500} = \sqrt{252500}

\approx 502.5 \text{ meters}

📖 NASA: Ballistic Flight Equations 📖 Khan Academy: Distance Formula

Frequently Asked Questions

The Pythagorean theorem states that in a right triangle, a² + b² = c², where a and b are the legs and c is the hypotenuse (the side opposite the right angle).

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