page_title: Pythagorean theorem
section_title: Hyperbolic geometry
hierarchical_section_title: Pythagorean theorem / Generalizations / Non-Euclidean geometry / Hyperbolic geometry
context_page_description: In mathematics, the Pythagorean theorem, also known as Pythagoras' theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse is equal to the sum of the areas of the squares on the other two sides. This theorem can be written as an equation relating the lengths of the sides a, b and c, often called the "Pythagorean equation":
  where c represents the length of the hypotenuse and a and b the lengths of the triangle's other two sides. The theorem, whose history is the subject of much debate, is named for the ancient Greek thinker Pythagoras.
  The theorem has been given numerous proofs – possibly the most for any mathematical theorem. They are very diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years. The theorem can be generalized in various ways, including higher-dimensional spaces, to spaces that are not Euclidean, to objects that are not right triangles, and indeed, to objects that are not triangles at all, but n-dimensional solids.
context_section_description: In a hyperbolic space with uniform curvature −1/R², for a right triangle with legs a, b, and hypotenuse c, the relation between the sides takes the form:
  where cosh is the hyperbolic cosine. This formula is a special form of the hyperbolic law of cosines that applies to all hyperbolic triangles:
  with γ the angle at the vertex opposite the side c.
  By using the Maclaurin series for the hyperbolic cosine, cosh x ≈ 1 + x²/2, it can be shown that as a hyperbolic triangle becomes very small (that is, as a, b, and c all approach zero), the hyperbolic relation for a right triangle approaches the form of Pythagoras's theorem.
  For small right triangles (a, b << R), the hyperbolic cosines can be eliminated to avoid loss of significance, giving
caption: Hyperbolic triangle