Área triângulo esférico
- Created by
- Renato Passos, Eng. de Software
- Reviewed by
- Renato Passos, Eng. de Software
Last updated: Apr 18, 2026
About this calculator
The spherical triangle area calculator computes the surface of a triangle formed by three arcs of great circles on a sphere. Unlike planar triangles, its calculation depends on the sphere's radius (R) and the sum of its internal angles (A, B, C). The formula used is R²·(A+B+C−π), where π accounts for the sphere's curvature.
This type of calculation is critical in air navigation, astronomy, and geography, where Earth or other spheres cannot be simplified as flat. For example, flight routes between cities follow great circle arcs to optimize distances.
NOTE: Angles must be provided in radians, and the sphere's radius should be in the same unit as the resulting area. Common errors include using degrees instead of radians or misapplying Earth's radius with approximate values.
The formula requires the sum of the triangle's angles to be greater than π (180°), a unique feature of spherical geometry. This occurs because the curvature of the surface alters the properties of triangles compared to Euclidean geometry.
Frequently asked questions
Why do angles need to be in radians?
The spherical triangle area formula requires angles in radians to maintain consistency with the sphere's radius unit. Degrees are incompatible with the formula's derivation.
How does this area differ from a planar triangle?
In spherical geometry, area depends on angular excess (A+B+C−π), while in planar geometry it depends only on base and height. The sphere's curvature directly alters the relationship between angles and area.
Can the calculator use Earth as a reference?
Yes, as long as you input Earth's radius (approximately 6,371 km) and the corresponding angles on Earth's surface, such as in flight paths or geographic navigation.
What unit should I use for the sphere's radius?
Use the same unit as your desired area. If you want the result in square kilometers, the radius must be in kilometers. Inconsistent units produce invalid results.