Pythagorean Theorem
Free Pythagorean Theorem for math. Enter values to solve step-by-step with verified formulas, worked examples, and detailed explanations. Free math tool.
Formula
a² + b² = c²
The square of the hypotenuse (c) equals the sum of squares of the legs (a and b).
Worked Examples
Example 1: Standard 3-4-5
Problem:a=3, b=4
Solution:c = √(3² + 4²) = √(9 + 16) = √25 = 5
Result:5
Example 2: Large Triangle
Problem:a=5, b=12
Solution:c = √(25 + 144) = √169 = 13
Result:13
Frequently Asked Questions
Does this theorem work for all triangles?
No, the Pythagorean Theorem ONLY applies to right-angled triangles (triangles containing one 90° angle).
What is the Hypotenuse?
The hypotenuse is the longest side of a right triangle. It is always located opposite the 90-degree (right) angle.
What are the "Legs" of a triangle?
The legs are the two shorter sides of a right triangle that meet to form the right angle.
What is a Pythagorean Triple?
A set of three whole numbers that satisfy the equation a² + b² = c². Common examples include 3-4-5, 5-12-13, and 8-15-17.
How do I find a missing leg?
If you know the hypotenuse (c) and one leg (b), use the formula a = √(c² - b²).
Is the 3-4-5 triangle special?
Yes, it is the simplest Pythagorean triple and the only one consisting of consecutive integers. It's often used in construction to check for square corners.
What is the Converse of the Pythagorean Theorem?
It 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.
What is the distance formula?
The distance formula between two points (x1,y1) and (x2,y2) is derived directly from the Pythagorean theorem: d = √((x2-x1)² + (y2-y1)²).
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 (a squared + b squared = c squared). It only applies to right triangles and is fundamental to geometry and trigonometry.
Background & Theory
The Pythagorean Theorem:
In a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
Formula:
a² + b² = c²
Where:
- a, b: The lengths of the legs (sides adjacent to right angle).
- c: The length of the hypotenuse (longest side).
Applications:
- Construction and carpentry (ensuring square corners).
- Navigation (calculating distances).
- Physics (vectors).