Pythagorean Theorem Calculator

Calculate the sides of a right triangle using the formula a² + b² = c².

Select which side you want to solve for, enter the known side lengths, and get the missing side instantly.

Practical Examples

Click on any example to load the data into the calculator and see how it works.

Find Hypotenuse 'c'

c

A classic 3-4-5 right triangle. Given legs a=3 and b=4, find the hypotenuse.

a = 3, b = 4

Find Leg 'a'

a

A common 5-12-13 right triangle. Given leg b=12 and hypotenuse c=13, find leg a.

b = 12, c = 13

Find Leg 'b'

b

An 8-15-17 right triangle. Given leg a=8 and hypotenuse c=17, find leg b.

a = 8, c = 17

Decimal Values

c

A right triangle with decimal lengths. Given a=2.5 and b=6, find the hypotenuse.

a = 2.5, b = 6

Other Titles
Understanding the Pythagorean Theorem: A Comprehensive Guide
An in-depth look at the famous theorem, its applications, and how to use this calculator effectively.

What is the Pythagorean Theorem?

  • The Core Formula: a² + b² = c²
  • Understanding the Parts of a Right Triangle
  • A Brief History of the Theorem
The Pythagorean Theorem is a fundamental principle in Euclidean geometry that describes the relationship between the three sides of a right-angled triangle. A right triangle is any triangle that has one angle equal to exactly 90 degrees.
The Formula
The theorem is stated as the algebraic equation: a² + b² = c². In this equation, 'a' and 'b' represent the lengths of the two shorter sides of the triangle, known as the 'legs'. The side 'c' represents the length of the longest side, which is called the 'hypotenuse'. The hypotenuse is always the side opposite the right angle.
This powerful formula allows you to find the length of a missing side if you know the lengths of the other two sides.

Common Pythagorean Triples

  • 3, 4, 5 (since 3² + 4² = 9 + 16 = 25 = 5²)
  • 5, 12, 13 (since 5² + 12² = 25 + 144 = 169 = 13²)
  • 8, 15, 17 (since 8² + 15² = 64 + 225 = 289 = 17²)

Step-by-Step Guide to Using the Calculator

  • Choosing Which Side to Solve For
  • Inputting Your Known Values
  • Interpreting the Results and Steps
Our calculator simplifies the process of applying the Pythagorean theorem. Here's how to use it:
1. Select the Side to Solve For
Start by using the 'Side to Solve For' dropdown menu. Choose whether you need to find the length of Leg 'a', Leg 'b', or the Hypotenuse 'c'.
2. Enter the Known Side Lengths
The calculator will show input fields for the two sides you need to know. For example, if you're solving for 'c', you'll need to enter values for 'a' and 'b'. Enter positive numbers in these fields.
3. Calculate and View Results
Click the 'Calculate' button. The tool will instantly display the length of the missing side, along with a detailed step-by-step breakdown of how the answer was derived.

Calculation Scenarios

  • Solving for Hypotenuse 'c': Formula used is c = √(a² + b²).
  • Solving for Leg 'a': Formula used is a = √(c² - b²). Note that c must be greater than b.
  • Solving for Leg 'b': Formula used is b = √(c² - a²). Note that c must be greater than a.

Real-World Applications of the Pythagorean Theorem

  • Architecture and Construction
  • Navigation and Surveying
  • Design and Graphics
The Pythagorean theorem is not just an academic exercise; it's used to solve practical problems every day.
Construction and Architecture
Builders and architects use the theorem to ensure that corners are perfectly square. By measuring two sides of a foundation or frame (e.g., 3 feet and 4 feet), they can check if the diagonal measurement is correct (5 feet) to confirm a true 90-degree angle.
Navigation
Pilots and ship captains can use the theorem to calculate the shortest distance between two points. If they travel a certain distance east and then a certain distance north, their straight-line distance from the starting point forms the hypotenuse of a right triangle.
Video Games and Graphics
In computer graphics, the distance between two objects (or pixels) on a 2D screen is often calculated using the Pythagorean theorem, which is essential for collision detection, character movement, and more.

Application Examples

  • A ladder leaning against a wall forms a right triangle. If the wall is 12 feet high and the base of the ladder is 5 feet from the wall, the ladder's length is 13 feet.
  • A TV screen's size is its diagonal. A 40-inch wide and 30-inch tall screen would be a 50-inch TV.

Common Misconceptions and Correct Methods

  • Applying to Non-Right Triangles
  • Mixing Up Legs and Hypotenuse
  • Forgetting to Take the Square Root
While powerful, the Pythagorean theorem is sometimes misapplied. Understanding these common errors can help ensure you get the correct result.
Mistake 1: Using it for Any Triangle
The most critical rule is that the Pythagorean theorem only applies to right-angled triangles. For other triangles (acute or obtuse), you must use other laws, like the Law of Sines or the Law of Cosines.
Mistake 2: Confusing 'c' with 'a' or 'b'
The hypotenuse 'c' is always the longest side and must be used alone on one side of the equation. When solving for a leg ('a' or 'b'), you must subtract its square from the hypotenuse's square (e.g., c² - b²), not the other way around. The hypotenuse must always be larger than either leg.
Mistake 3: Forgetting the Final Step
A common oversight is to calculate a² + b² and forget to take the square root to find 'c'. Remember, the formula gives you c², not 'c'.

Correction Examples

  • Incorrect: a=3, c=5. Calculating b = √(3² + 5²) is wrong.
  • Correct: a=3, c=5. Calculating b = √(5² - 3²) = √(25 - 9) = √16 = 4.

Mathematical Derivation and Proofs

  • Euclid's Geometric Proof by Rearrangement
  • An Algebraic Proof
  • The Relationship to Distance Formula
There are hundreds of published proofs of the Pythagorean theorem. Most fall into two categories: geometric proofs (using areas) and algebraic proofs.
Proof by Rearrangement
One of the most intuitive proofs involves a large square with a side length of (a+b). Inside this square, you can place four identical right triangles (with sides a, b, c) and a smaller square with side 'c'. The total area can be expressed in two ways: as (a+b)² and as the sum of the areas of the four triangles (4 1/2 a * b) plus the area of the inner square (c²). Setting these equal and simplifying proves that a² + b² = c².
The Distance Formula
The Pythagorean theorem is also the foundation for the distance formula in coordinate geometry. The distance 'd' between two points (x₁, y₁) and (x₂, y₂) is given by d = √((x₂-x₁)² + (y₂-y₁)²), which is a direct application of the theorem where (x₂-x₁) and (y₂-y₁) are the 'a' and 'b' sides of the triangle.

Proof-related Concepts

  • Area of outer square: (a+b)² = a² + 2ab + b²
  • Area of inner parts: 4(½ab) + c² = 2ab + c²
  • Equating them: a² + 2ab + b² = 2ab + c² => a² + b² = c²