Calculations at a spherical triangle or Euler triangle. The spherical triangle doesn't belong to the Euclidean, but to the spherical geometry. The three sides are parts of great circles, every angle is smaller than 180°, all together are larger than 180°.
Enter radius of the sphere and three angles and choose the number of decimal places. Then click Calculate. Please enter angles in degrees, here you can convert angle units.
Formulas:
pi:
Radius is a one-dimensional unit (e.g. meter), the area has this unit squared (e.g. square meter). The sperical excess is an angle, calculated in degrees.
Euclidean geometry is the geometry of the plane and of flat space. Until the beginning of the 19th century, it was regarded as the only geometry. Certain rules apply within it, such as the invariant sum of angles in polygons with the same number of vertices, like 180 degrees for a triangle and 360 degrees for a quadrilateral. Also the parallel postulate applies. On a plane, for any given line and a point not lying on it, there exists exactly one other line passing through that point which does not intersect the first line and is parallel to it. In the course of their futile attempts to prove this postulate, János Bolyai, Nikolai Lobachevsky and Carl Friedrich Gauß discovered geometries in which this rule does not hold, the non-Euclidean geometries.
One such non-Euclidean geometry is spherical geometry, another is hyperbolic geometry. A triangle drawn on the surface of a sphere does not have an angle sum of 180 degrees. In fact, for any three points on a sphere that do not lie on a common great circle, there exist multiple possible triangles, usually eight. The term spherical triangle typically, as also here, refers to the Eulerian spherical triangle, which is the smallest of these triangles. In this specific triangle, every angle measures less than 180 degrees, a characteristic that does not apply to the other possible triangles on the sphere.
When measuring distances and angles on Earth, beyond a certain length, this is done using spherical geometry. Although the Earth is not exactly a sphere, but rather an oblate spheroid with bumps and dents, a spherical model is often sufficient as an approximation.