The Pythagorean Theorem on a Sphere and in Non-Euclidean Geometry

No — on a sphere, a² + b² = c² is not exactly true. For a right triangle drawn on a sphere of radius R, with sides measured along great circles, the exact rule is cos(c/R) = cos(a/R)cos(b/R). The hypotenuse comes out shorter than the flat formula predicts, but for triangles much smaller than the sphere the difference is negligible.

On the hyperbolic plane, the other classical non-Euclidean geometry, the cosines become hyperbolic cosines and the hypotenuse comes out longer. This page derives the spherical version, shows why both versions collapse to the ordinary Pythagorean theorem for small triangles, and explains why the flat theorem depends on Euclid's parallel postulate.

Why the flat theorem needs a flat world

Euclid built geometry on five postulates. The fifth, the parallel postulate, is equivalent to the familiar statement that through a point not on a line there is exactly one parallel to that line. For two thousand years mathematicians tried to prove it from the other four and failed. In the 19th century Nikolai Lobachevsky (who published in 1829) and János Bolyai (around 1832) showed why: a perfectly consistent geometry exists in which the postulate is false. Gauss had reached the same conclusion privately but never published it.

The Pythagorean theorem, which is far older than Euclid (see its history), is tied to that postulate. Every proof of it uses the parallel postulate somewhere, often invisibly — Euclid's proof needs squares, and a four-sided figure with four right angles exists only if the parallel postulate holds. In fact, on top of Euclid's other axioms, the Pythagorean theorem and the parallel postulate are logically equivalent: each implies the other. So once you leave the flat plane, the theorem must change.

Right triangles on a sphere

On a sphere the role of straight lines is played by great circles — circles whose center is the center of the sphere, such as the equator or any line of longitude. The shortest path between two points on the surface runs along a great circle, which is why long-haul flights follow curved-looking routes on flat maps. A spherical triangle has three great-circle arcs as sides, and its side lengths are measured along the surface.

Spherical triangles behave strangely. Their angles always add up to more than 180°, and the excess grows with the triangle's area. The extreme example is easy to picture on Earth.

Start at the North Pole, walk south along the prime meridian to the equator (a point in the Gulf of Guinea), turn left and walk east along the equator to longitude 90° E (in the Indian Ocean, west of Sumatra), then turn left again and walk back north to the pole. Each leg is a quarter of a great circle, and every turn is exactly 90°.

Take Earth as a sphere of mean radius R ≈ 6,371 km. Each side is πR over 2 ≈ 10,008 km. Treat NA and AB as legs of a right triangle and the flat formula predicts NB = √2 × 10,008 ≈ 14,153 km. The real distance is 10,008 km. Here the flat theorem is not slightly off; it is off by more than 4,000 km.

This triangle also illustrates Girard's theorem: the area of a spherical triangle equals R² times its angle excess (angle sum minus 180°, in radians). Here the excess is 90°, or π over 2, so the area is π over 2R² — exactly one-eighth of Earth's surface, about 63.8 million km².

The spherical Pythagorean theorem

For a right triangle on a sphere of radius R, with legs a and b and hypotenuse c measured along the surface:

The quantities a/R, b/R and c/R are the angles (in radians) that each side subtends at the center of the sphere. Check it on the three-right-angle triangle: cosπ over 2 · cosπ over 2 = 0 = cosπ over 2, so c = πR over 2, as it should be.

A short proof with vectors

Put the sphere's center at the origin. For any two points P and Q on the sphere, the dot product of their position vectors is P · Q = R²cosθ, where θ is the angle between them — and the surface distance from P to Q is Rθ.

1. Place the right angle. Put the right-angle vertex at C = (R, 0, 0). Run one leg along the great circle in the xy-plane and the other along the great circle in the xz-plane. Their directions at C are the y- and z-axes, which are perpendicular, so the angle at C is 90°.
2. Write the other two corners. Moving a distance b along the first circle and a along the second gives A = R(cosb over R,  sinb over R,  0) and B = R(cosa over R,  0,  sina over R). Both really lie on the sphere, because cos²θ + sin²θ = 1 (the Pythagorean identity).
3. Take the dot product. A · B = R²cosa over Rcosb over R + 0 + 0. But also A · B = R²cosc over R, because c is the distance from A to B. Setting the two equal gives the theorem.

The formula is the special case C = 90° of the spherical law of cosines, cosc over R = cosa over Rcosb over R + sina over Rsinb over RcosC, just as the flat theorem is the special case of the law of cosines.

Small triangles: how a² + b² = c² comes back

Surveyors use the ordinary theorem on a round planet without any trouble, and a series expansion shows why. Expand the cosines in their Taylor series, cosx = 1 − x² over 2 + x⁴ over 24 − ⋯, and substitute into the spherical formula:

The 1s cancel. Multiply everything by −2R² and keep the largest terms. To leading order the equation says c² = a² + b² — the flat theorem. Keeping the next terms as well (and using c⁴ ≈ (a² + b²)² inside them) gives the first correction:

The correction is tiny when the legs are small compared with R, because it is divided by R². As R → ∞ — a sphere so large it is effectively flat — it disappears completely.

Plug in Earth-sized numbers to see how small "small" is. For an isosceles right triangle with legs L, the correction term says the flat formula overestimates the hypotenuse by about L³ over 6√2 R². The table uses the exact spherical formula:

That is why builders, surveyors and video-game engines use the flat theorem without a second thought, while airline navigators and GPS software switch to spherical (or ellipsoidal) formulas; see the Pythagorean theorem in navigation and the flat-map distance formula.

The hyperbolic Pythagorean theorem

Hyperbolic geometry is the geometry of Lobachevsky and Bolyai: through a point off a line there are infinitely many parallels, triangles have angle sums less than 180°, and space "spreads out" faster than a plane, like the frilly edge of a lettuce leaf or a saddle that is saddle-shaped everywhere. Its curvature is negative, −1/R². For a right triangle with legs a, b and hypotenuse c:

Here coshx = 1 over 2(e^x + e^−x) is the hyperbolic cosine, whose series is 1 + x² over 2 + x⁴ over 24 + ⋯ — the cosine series with every sign made positive. The same expansion as before gives

so hyperbolic hypotenuses are slightly longer than flat ones, and again the ordinary theorem returns for small triangles. The vector proof carries over too: in the "hyperboloid model" of the hyperbolic plane the dot product is replaced by x₁y₁ + x₂y₂ − x₃y₃, and the same three steps turn every cos into cosh.

Pythagoras survives in the small: the line element

The small-triangle limit is not a coincidence; it is built into how modern geometry measures curved spaces. In Riemannian geometry (and in Einstein's general relativity) distance is defined by a line element that says how long a tiny step is. In a flat plane with ordinary coordinates it is the Pythagorean theorem for an infinitesimal right triangle:

On a sphere, using latitude φ and longitude λ, it becomes ds² = R² dφ² + R²cos²φ dλ² — still a sum of squares of two perpendicular steps, but with weights that change from place to place. Every curved surface looks flat if you zoom in far enough, and at that scale the Pythagorean theorem is exactly right. Curvature is what you see when the small right triangles are stitched together into large ones. For how this plays out in physics, see the Pythagorean theorem in physics.