Pythagorean Theorem Calculator (Right Triangle Solver)
Find any missing side of a right triangle with the Pythagorean theorem, plus area, perimeter, and both acute angles with step-by-step substitutions.
Pythagorean Calc — interactive calculator, computed instantly in your browser.
The Pythagorean Theorem: Solving Right Triangles Step by Step
3² + 4² = 5² — The Classic Right Triangle
The Pythagorean theorem — a² + b² = c² — states that the two legs of a right triangle add up to the square of its hypotenuse. It is the oldest working formula in geometry (Babylonian tablets predate Pythagoras by a millennium) and still the fastest way to get a distance: from laying out a roof truss to the GPS ping calculating how far away you are.
1. Identify the Sides, Then Apply the Formula
- c (hypotenuse) is always the longest side, opposite the right angle — swap it with a leg and the numbers break
- Memorized triples make checking instant: 3-4-5, 5-12-13, 8-15-17, 7-24-25 — any multiple (6-8-10,15-20-25) works too
- Carpenters use the3-4-5 rule to square a foundation: measure3 ft one way,4 ft the other, and the diagonal must be exactly5 ft
- Non-triple answers are irrational: legs7 and24 give hypotenuse25, but7 and9 give √130 = 11.4018…
2. Solving for a Missing Leg
Rearrange before computing — subtract instead of add: given hypotenuse13 and leg5, the missing leg is √(13² − 5²) = √144 = 12. Because subtraction shrinks the radicand, a leg can never exceed the hypotenuse: try √(5² − 13²) and you get a negative under the root — proof those sides cannot form a right triangle. The calculator flags exactly this case instead of printing NaN.
3. Where the Theorem Shows Up
- Distances on a coordinate plane: Δx and Δy are legs — the distance formula in the slope calculator is this theorem (see √(Δx² + Δy²))
- Screen diagonals: width and height are legs; a27-inch16:9 panel measures its diagonal by the same sum of squares
- Construction & fabrication: stair stringers, rafter cuts, ramp slopes — verify right angles before cutting material
- From the completed sides the calculator also derives area (a × b ÷2), perimeter, and both acute angles via arctangent — the full right-triangle toolkit in one pass
🔗 Geometry & Algebra Cross-References
The theorem is coordinate geometry's workhorse — pair it with our 🧮 Math Hub for slope, quadratic, and exponent solvers.
Sides become coordinates: once you have two points, the slope calculator returns rise, run, and segment length for the same triangle.
Frequently Asked Questions (Pythagorean Calc)
For any right triangle, a² + b² = c² where a and b are the legs and c is the hypotenuse. Rearranged: c = √(a² + b²) for the hypotenuse, or a = √(c² − b²) for a missing leg — subtraction always isolates the unknown side.
Square the legs, add, then take the root: 3² + 4² = 9 + 16 = 25, and √25 = 5. The3-4-5 triple is the most common whole-number right triangle, which is why builders carry it in their heads for quick field checks.
Then the triangle is not right-angled — it is either obtuse (a² + b² less than c²) or acute (a² + b² greater than c²). This calculator reports the mismatch so you can re-check which side you assumed was the hypotenuse; it must be the longest of the three.
Anywhere distance hides behind perpendicular components: navigation and GPS offsets, screen diagonal sizes, roofing and framing squareness, ramp accessibility checks, and machine design — plus every coordinate-geometry distance formula derived from it.