Pythagorean Theorem Calculator
Solve for any side of a right triangle, verify whether three lengths form a right angle, or work out the distance between two coordinate points — with simplified radicals and the classic squares-on-each-side proof drawn to scale.
⚡ Quick Tips for Using This Calculator Correctly
Features
Hypotenuse or either leg, plus area, perimeter, altitude to the hypotenuse and both acute angles.
Enter any three sides and instantly learn whether the triangle is right, acute or obtuse.
Straight-line distance between two (x, y) points, using the same underlying theorem.
Exact answers like 3√5 shown alongside the decimal, not just an approximation.
The classic geometric proof drawn to true scale — visually confirms a² + b² = c².
Quick-fill buttons for 3-4-5, 5-12-13 and four other common whole-number triples.
One Equation, Three Different Questions
Most free Pythagorean calculators do exactly one thing: given two legs, find the hypotenuse. That covers the textbook case but misses two questions that come up just as often in practice — "do these three measurements actually form a right angle?" and "how far apart are these two points?" Both are the same underlying equation, a² + b² = c², just applied to a different situation. This tool treats all three as first-class modes rather than forcing everything through a single hypotenuse-only calculator.
Take the classic 3-4-5 triangle: legs of 3 and 4 give a hypotenuse of exactly 5, since 9 + 16 = 25. That's Find a Side mode. Now flip the question: someone measures a corner at 3 feet, 4 feet and 5 feet along the diagonal, and needs to confirm the corner is actually square before pouring a foundation. That's Verify Triangle mode — checking that 3² + 4² lands exactly on 5² rather than close to it. And if those same three points were instead coordinates on a site plan, Distance Formula mode measures the gap between any two of them directly.
Why the Squares-on-Sides Diagram Isn't Just Decoration
The proof usually shown in geometry class draws a square built outward on each of the triangle's three sides — one sized a×a, one sized b×b, one sized c×c — and demonstrates visually that the two smaller squares' combined area always equals the larger square's area. This calculator draws that exact construction to true scale from whatever values are entered, rather than a fixed generic illustration. Watching the a² and b² squares visually "add up to" the c² square makes the theorem tangible in a way a results table alone doesn't, and it makes an input mistake — a leg and the hypotenuse accidentally swapped — immediately obvious, since the squares stop lining up correctly.
Verify Triangle mode reuses the same drawing whenever the three entered sides genuinely form a right angle, which turns an abstract pass/fail result into something visibly checkable. If the numbers describe an obtuse or acute triangle instead, the diagram is skipped rather than forced onto a shape it was never meant to represent — a small detail, but one that keeps the visual honest instead of decorative.
Where Pythagorean Triples Actually Come From
The quick-fill buttons above load six whole-number triples — (3,4,5), (5,12,13), (8,15,17), (7,24,25), (20,21,29) and (9,40,41) — but they aren't a random list. Every one of them is generated by the same formula: pick two positive integers m > n, and a = m²−n², b = 2mn, c = m²+n² always satisfies a² + b² = c² exactly. For m=2, n=1: a=3, b=4, c=5. For m=3, n=2: a=5, b=12, c=13. Push m and n further and the triples keep coming — there are infinitely many, and no two share the same shape once scaled to their smallest whole-number form.
A triple counts as primitive when its three numbers share no common factor — (3,4,5) is primitive, while (6,8,10) is just that same triple doubled, and (9,12,15) is it tripled. Every non-primitive triple is a whole-number multiple of exactly one primitive triple, which is why construction trades default to the smallest version: a 3-4-5 corner check needs the shortest possible tape measurements to confirm the same right angle a 30-40-50 check would. To confirm a set of three numbers truly shares no common factor before trusting it as primitive, the LCM & GCD Calculator settles it in one step.
Who Actually Needs Right-Triangle Math
Students checking geometry homework across all three question types, not just the find-the-hypotenuse case most textbooks lead with. Carpenters and builders squaring corners with the ancient 3-4-5 rope trick — measure 3 feet one way, 4 feet the other, and the diagonal must read exactly 5 feet for the corner to be a true right angle. Surveyors and landscapers verifying that a staked-out plot's corners are genuinely square before any permanent work begins. Programmers and game developers computing on-screen or in-world distances between coordinates, which is precisely what Distance Formula mode replicates. For triangles that aren't right-angled at all — three sides with no guaranteed 90° corner — the Triangle Calculator handles the general case using the Law of Cosines and Law of Sines.
Two Calculations Worth Walking Through
Squaring a foundation corner: a builder measures 6 feet along one wall line and 8 feet along the perpendicular wall line, then checks the diagonal. Find a Side mode gives c = √(6² + 8²) = 10 feet exactly — a clean whole number because (6, 8, 10) is just the (3, 4, 5) triple doubled. If the measured diagonal comes out to 10 feet 2 inches instead, Verify Triangle mode confirms the corner is slightly off-square rather than leaving that judgment to eyeballing a tape measure.
Screen-distance check: two UI elements sit at pixel coordinates (1, 2) and (4, 6) on a layout grid. Distance Formula mode computes Δx = 3, Δy = 4, and a straight-line distance of exactly 5 pixels — again the 3-4-5 triple, just measured in screen space instead of physical feet, which is exactly how collision detection and layout-spacing checks work under the hood in most rendering code.
Why Browser-Based Beats a Framing Square and Calculator App
A framing square confirms a right angle physically but can't tell you by how much a slightly-off corner actually misses square, and a basic phone calculator won't simplify a radical or draw the proof. This tool runs every formula and renders the diagram entirely client-side in JavaScript, with nothing entered ever sent to a server — safe for proprietary site plans or homework kept private. For angle-heavy work beyond a single right triangle — arbitrary sine, cosine and tangent values — the Scientific Calculator covers that directly.
Where This Calculator Draws the Line
It solves single right triangles, verifies triangle type from three sides, and computes 2D coordinate distance — it doesn't extend to 3D distance between points in space, oblique (non-right) triangles, or trigonometric identities beyond a single right angle. For a circle's own geometry — sector, segment or ring area — the Circle Calculator picks up where this one intentionally stops.