Last updated: 2026-05-13

The Pythagorean Theorem

The Pythagorean theorem states that in any right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides:

$a^2 + b^2 = c^2$

It is one of the oldest results in mathematics. Babylonian clay tablets from around 1800 BCE already list Pythagorean triples — sets of integers satisfying the equation — centuries before Pythagoras was born. What Pythagoras and his followers contributed was a general proof: the relationship holds for every right triangle, not just specific integer examples.

Using this calculator

Enter any two sides of a right triangle and the calculator solves for the third. All three fields are editable: fixing the hypotenuse and one leg finds the remaining leg, and providing both legs gives the hypotenuse.

• a and b known → find c (hypotenuse): $c = \sqrt{a^2 + b^2}$
• b and c known → find a: $a = \sqrt{c^2 - b^2}$
• a and c known → find b: $b = \sqrt{c^2 - a^2}$

All length units are supported, and metric and imperial units can be mixed freely.

Why the formula works

The most intuitive proof draws three squares on the sides of a right triangle. The area of the square on the hypotenuse equals the combined area of the squares on the two legs, which can be confirmed by rearranging the pieces to cover the larger square exactly. This "dissection proof" requires no algebra, only area counting, which is why it has been rediscovered across cultures throughout history.

The algebraic statement follows from the distance formula in coordinate geometry. Placing the right angle at the origin, the legs along the axes, and the hypotenuse as the line segment from $(a, 0)$ to $(0, b)$, its length is $\sqrt{a^2 + b^2}$ by definition of Euclidean distance.

Pythagorean triples

Pythagorean triples are integer solutions to $a^2 + b^2 = c^2$. A few worth knowing:

a	b	c
3	4	5
5	12	13
8	15	17
7	24	25

Any positive integer multiple of a triple is another triple. So (6, 8, 10), (9, 12, 15), and (15, 20, 25) are all derived from the (3, 4, 5) family.

The general formula for generating all primitive triples (those sharing no common factor) uses two integers $m > n > 0$:

$a = m^2 - n^2, \quad b = 2mn, \quad c = m^2 + n^2$

For $m = 2, n = 1$: $a = 3, b = 4, c = 5$ — the classic.

Real-world applications

Construction and carpentry. The 3-4-5 rule is a common method for checking right angles without a protractor. Marking 3 units along one wall and 4 units along the adjacent wall, then measuring the diagonal between the endpoints, confirms a square corner when the diagonal is exactly 5 units.

Navigation. The straight-line distance between two points on a flat map is the hypotenuse of a right triangle whose legs are the north-south and east-west separations. GPS receivers and mapping software use this constantly.

Physics and engineering. Vector magnitudes in two dimensions are computed via the Pythagorean theorem. The speed of an object with horizontal velocity $v_x$ and vertical velocity $v_y$ is $\sqrt{v_x^2 + v_y^2}$.

Computer graphics. Distance between pixels, collision detection, and rendering calculations all rely on the theorem. The 3D extension — $d = \sqrt{x^2 + y^2 + z^2}$ — is direct generalization.

Limited to right triangles

The Pythagorean theorem holds only when one angle is exactly 90°. For any other triangle, the correct tool is the law of cosines:

$c^2 = a^2 + b^2 - 2ab\cos(C)$

where $C$ is the angle opposite side $c$. When $C = 90°$, $\cos(90°) = 0$ and the law of cosines reduces exactly to the Pythagorean theorem — confirming that it is a special case of a more general result.

Frequently Asked Questions (FAQ)

What are Pythagorean triples?

Pythagorean triples are sets of three positive integers that satisfy a² + b² = c². The most famous is (3, 4, 5): 9 + 16 = 25. Other common triples include (5, 12, 13), (8, 15, 17), and (7, 24, 25). Any multiple of a triple is also a triple — e.g. (6, 8, 10).

How is the Pythagorean theorem used in real life?

The theorem is used in construction (checking if corners are square), navigation (computing straight-line distance), physics (vector magnitudes), and computer graphics (2D and 3D distance calculations). Carpenters use the 3-4-5 rule to lay out right angles.

Does the Pythagorean theorem work for all triangles?

No — it only applies to right triangles (triangles with one 90° angle). For other triangles, use the law of cosines: c² = a² + b² − 2ab cos(C), where C is the angle opposite side c.