Pythagorean Theorem
Find the hypotenuse or a missing leg of a right triangle.
Hypotenuse (c)
5
c = √(a² + b²)
Side a
3
Side b
4
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How the Pythagorean Theorem works
The Pythagorean Theorem calculator lets you instantly find the hypotenuse or either missing leg of a right triangle by entering just two known side lengths. It's an essential tool for students, engineers, architects, carpenters, and anyone working with right-angle geometry in the real world.
At its core, this calculator applies the Pythagorean theorem — the foundational relationship a² + b² = c² — where 'a' and 'b' are the two legs of a right triangle and 'c' is the hypotenuse (the longest side, always opposite the right angle). You input any two of the three side values, and the calculator solves algebraically for the missing one. If you need the hypotenuse, it computes c = √(a² + b²). If a leg is missing, it rearranges to a = √(c² − b²) or b = √(c² − a²).
The key factor that makes this theorem work — and makes it exclusively applicable — is the right angle. The triangle must contain a 90° angle for the a² + b² = c² relationship to hold. Oblique triangles (those without a right angle) require different laws, such as the Law of Cosines. When using this calculator, always confirm your triangle is a right triangle; a common mistake is applying the Pythagorean theorem to a triangle that is merely assumed to be right-angled without verification, leading to incorrect results.
One practical consideration is unit consistency. The calculator works with any unit of length — inches, feet, meters, centimeters — but all three sides must be expressed in the same unit before you input values. For example, if one leg is measured in feet and another in inches, convert them to a common unit first. The result will be in whatever unit you used for your inputs. This is one of the most frequent sources of real-world errors, particularly in construction and DIY projects.
Beyond simple side-finding, the Pythagorean theorem underpins distance calculations in coordinate geometry (the distance formula is a direct derivation), 3D diagonal problems, and even checks for right angles in construction (the classic 3-4-5 rule). Recognizing Pythagorean triples — integer sets like 3-4-5, 5-12-13, and 8-15-17 — can serve as quick mental checks on your results. If your computed sides don't approximate a known triple or ratio, double-check your inputs for transcription errors or unit mismatches.
Formula
a² + b² = c²
Pro tips
- Always verify your triangle has a true right angle before using the calculator — applying a² + b² = c² to a non-right triangle will produce a mathematically clean but physically wrong answer.
- Use Pythagorean triples (3-4-5, 5-12-13, 8-15-17) as instant sanity checks; if your sides scale to one of these ratios, your answer is almost certainly correct.
- Convert all measurements to the same unit before entering values — mixing feet and inches or meters and centimeters is the single most common source of real-world calculation errors.
- When working in construction or carpentry, apply the 3-4-5 method in reverse: measure 3 ft along one wall, 4 ft along the adjacent wall, and confirm the diagonal is exactly 5 ft to verify a true 90° corner before building.
- For 3D diagonal problems (e.g., the longest rod that fits in a rectangular box), apply the theorem twice: first find the diagonal of the base (√(l² + w²)), then use that result as one leg with the height to find the full 3D diagonal.
Key terms
- Hypotenuse
- — The longest side of a right triangle, situated directly opposite the 90° angle, represented as 'c' in the Pythagorean theorem.
- Leg (Cathetus)
- — Either of the two shorter sides of a right triangle that form the right angle, represented as 'a' and 'b' in the formula.
- Right Triangle
- — A triangle containing exactly one 90° angle, which is the required condition for the Pythagorean theorem to apply.
- Pythagorean Triple
- — A set of three positive integers (e.g., 3-4-5 or 5-12-13) that satisfy a² + b² = c² exactly, producing a perfect right triangle with no irrational side lengths.
- Square Root
- — The inverse operation of squaring a number, used to isolate the unknown side after summing the squares of the two known sides.
- Distance Formula
- — A coordinate-geometry formula derived directly from the Pythagorean theorem that calculates the straight-line distance between two points: d = √((x₂−x₁)² + (y₂−y₁)²).
