Geometry essential
Pythagorean Theorem
The complete guide to a² + b² = c², what it means, how to prove it, how to use it, and where it appears in the real world.
Definition
What Is the Pythagorean Theorem?
The Pythagorean theorem is a fundamental rule in geometry that describes the relationship between the three sides of a right triangle. It states that the square of the hypotenuse, the side opposite the right angle, is equal to the sum of the squares of the other two sides, called the legs.
a² + b² = c²
Written as an equation: a² + b² = c², where a and b are the
lengths of the two legs, and c is the length of the hypotenuse. This equation holds
true for every right triangle, regardless of size or orientation.
The theorem applies only to right triangles, triangles that contain exactly one 90-degree angle. For triangles with no right angle, different rules apply. The Law of Cosines handles the general case. The right angle is usually marked with a small square in diagrams.
Formula
The Three Forms of the Formula
The basic equation a² + b² = c² can be rearranged to solve for any one of the three
sides. These three forms cover every possible right triangle calculation.
Use the Right Triangle Calculator to apply any of these formulas instantly with step-by-step working.
Proof
Geometric Proof of the Pythagorean Theorem
The most intuitive proof of the Pythagorean theorem is geometric: draw a square on each side of a right triangle, and the area of the square on the hypotenuse exactly equals the combined area of the squares on the two legs. The interactive diagram below lets you see this relationship directly.
What the proof animation shows
- The purple and pink squares represent
a²andb². - The green square represents
c², the area on the hypotenuse. - When the triangle is right, the two smaller areas add to the larger area.
- Because the relationship is about area, it works for every right triangle, not one diagram.
How to read it
- Start with the triangle alone and identify the right angle.
- Open the proof view to place one square on each side.
- Compare the colored areas instead of the side lengths directly.
- Read the equation as an area balance:
a² + b² = c².
This is one of hundreds of known proofs. Other approaches include algebraic rearrangement, similar triangles, and the dissection method. For a full walkthrough of multiple proof methods, see the Pythagorean Theorem Proof page.
Examples
Worked Examples
These two examples show how to apply the formula in both directions, finding the hypotenuse and finding a missing leg.
3-4-5 triangle
The classic first example: with legs 3 and 4, the hypotenuse is 5.
c = √(3² + 4²) = 5
Basic
Ladder against a wall
Use the diagonal ladder length and wall height to recover the ground distance.
x = √(10² - 8²) = 6
Basic
For more examples including word problems and special triangles, visit Pythagorean Theorem Examples or Word Problems with Solutions.
History
History and Origins
The theorem is named after Pythagoras of Samos, a Greek mathematician and philosopher who lived around 570–495 BCE. His school, the Pythagoreans, is credited with providing one of the earliest formal proofs of the relationship. However, the underlying idea was known long before Pythagoras.
Babylonian clay tablets from around 1800 BCE, most famously Plimpton 322, list sets of numbers
that satisfy a² + b² = c², suggesting that Babylonian scribes understood the
relationship in a practical sense. Ancient Indian texts, the Sulba Sutras, and Chinese
mathematical works such as the Zhoubi Suanjing contain similar results, applied to construction
and surveying.
What made the Greek contribution distinctive was the move toward formal, deductive proof, the idea that the theorem must be true for all right triangles, not just specific examples. Euclid's Elements contains a rigorous proof that remains a model of geometric reasoning. Today the theorem bears Pythagoras's name by convention, even though the history is far broader than one person.
Applications
Real-Life Applications
The Pythagorean theorem is not just a classroom exercise. Any time two measurements are perpendicular and you need the direct distance between two points, the theorem applies.
Construction
Builders verify square corners by checking that the diagonal of a rectangle matches √(l² + w²). The 3-4-5 method is a classic on-site technique.
Navigation
Pilots and sailors calculate straight-line distance from perpendicular displacement components, east-west and north-south offsets.
Screen Sizes
Display manufacturers specify screen size as the diagonal, calculated from width and height using the theorem.
Sports
Coaches calculate the diagonal distance across a field or court to plan drills and passing routes.
Game Development
Collision detection and movement vectors use distance formulas derived directly from the theorem.
Architecture
Roof pitch, staircase design, and ramp gradients all involve right triangle calculations.
Explore each application in depth at Pythagorean Theorem in Real Life.
Converse
The Converse of the Pythagorean Theorem
The converse of the Pythagorean theorem states the reverse: if the three sides of a triangle
satisfy a² + b² = c², then the triangle must be a right triangle. This gives you a
way to verify right angles without measuring them directly, useful in construction, carpentry, and
surveying.
For example, if a triangle has sides 5, 12, and 13, check whether 5² + 12² = 13²:
25 + 144 = 169. It does, so the triangle is a right triangle. If the equation does not hold, the
triangle is either acute, sum greater than c², or obtuse, sum less than
c².
For a full explanation with examples, see Converse of the Pythagorean Theorem.
Calculators
Pythagorean Theorem Calculator
Pick the tool that matches your problem. Each calculator focuses on one job and shows the full working.
Find the Hypotenuse
Enter two legs and get the hypotenuse with full working.
Use CalculatorFind a Missing Side
Enter the hypotenuse and one leg to find the other.
Use CalculatorSolve Any Right Triangle
Find all sides, angles, area, and perimeter.
Use CalculatorFAQ
Frequently Asked Questions
The Pythagorean theorem states that in any right triangle, the square of the hypotenuse
equals the sum of the squares of the two legs: a² + b² = c². It is one of the
most fundamental results in geometry and applies to every right triangle regardless of size.
The formula is a² + b² = c², where a and b are the two
legs and c is the hypotenuse. Rearranged, it gives
c = √(a² + b²) to find the hypotenuse, or
a = √(c² − b²) to find a missing leg.
No. It only applies to right triangles, triangles with exactly one 90-degree angle. For other
triangles, the Law of Cosines is the general rule:
c² = a² + b² − 2ab·cos(C). The Pythagorean theorem is a special case of that
formula when angle C is 90 degrees.
The theorem is named after Pythagoras of Samos, around 570–495 BCE, but the relationship was known to Babylonian, Indian, and Chinese mathematicians centuries earlier. Pythagoras or his school is credited with providing a formal deductive proof.
Pythagorean triples are sets of three positive integers that satisfy
a² + b² = c². The most well-known example is 3-4-5. Others include 5-12-13,
8-15-17, and 7-24-25. Multiples of any triple also work, such as 6-8-10.
It appears wherever two perpendicular measurements define a right triangle and you need the direct distance. Common uses include checking square corners in construction, calculating screen diagonals, estimating navigation distances, designing ramps and staircases, and computing movement vectors in game development.
The converse states that if a² + b² = c² holds for the three sides of a
triangle, then the triangle must be a right triangle. This lets you verify a right angle by
measuring sides alone. If a² + b² > c², the triangle is acute; if
a² + b² < c², it is obtuse.
Next steps
Explore Further
Calculators
Guides
Pythagorean Theorem Proof
Multiple proof methods explained step by step.
Pythagorean Theorem Examples
Worked examples from basic to advanced.
Word Problems with Solutions
Real-world problems solved step by step.
Converse of the Pythagorean Theorem
How to verify a right triangle from its sides.
Pythagorean Theorem in Real Life
Eight everyday applications with compact worked calculations.
How to Find the Hypotenuse
A step-by-step guide to using c = √(a² + b²).
Pythagorean Triples
Explore integer triples that satisfy a² + b² = c².