section_path: Miscellaneous triangles/Triangle in non-planar space
section_name: Triangle in non-planar space
paragraph_text: Triangle in non-planar space
  
  A non-planar triangle is a triangle not included in Euclidean space, roughly speaking a flat space. This means triangles may also be discovered in several spaces, as in hyperbolic space and spherical geometry. A triangle in hyperbolic space is called a hyperbolic triangle, and it can be obtained by drawing on a negatively curved surface, such as a saddle surface. Likewise, a triangle in spherical geometry is called a spherical triangle, and it can be obtained by drawing on a positively curved surface such as a sphere.
  
  The triangles in both spaces have properties different from the triangles in Euclidean space. For example, as mentioned above, the internal angles of a triangle in Euclidean space always add up to 180°. However, the sum of the internal angles of a hyperbolic triangle is less than 180°, and for any spherical triangle, the sum is more than 180°. In particular, it is possible to draw a triangle on a sphere such that the measure of each of its internal angles equals 90°, adding up to a total of 270°. By Girard's theorem, the sum of the angles of a triangle on a sphere is 180 ∘ × (1 + 4 f) {\displaystyle 180^{\circ }\times (1+4f)}, where f {\displaystyle f} is the fraction of the sphere's area enclosed by the triangle.
  
  In more general spaces, there are comparison theorems relating the properties of a triangle in the space to properties of a corresponding triangle in a model space like hyperbolic or elliptic space. For example, a CAT(k) space is characterized by such comparisons.