Area of Spherical Triangle

Theorem

Let $T$ be a spherical triangle on the surface of a sphere $S$.

The area $\AA$ of $T$ is given by:

$\AA = \dfrac {\pi r^2 E} {180}$

where:

$r$ is the radius of $S$
$E$ is the spherical excess of $T$.


Proof


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Sources

  • 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 4$: Geometric Formulas: Spherical Triangle of Angles $A, B, C$ on Sphere of Radius $r$: $4.44$
  • 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): spherical polygon
  • 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): spherical polygon
  • 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 7$: Geometric Formulas: Spherical Triangle of Angles $A, B, C$ on Sphere of Radius $r$: $7.44.$
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