Spherical Excess Calculator

How it is calculated

Enter the three interior angles A, B, C in degrees and the sphere radius R. The spherical excess is E = A + B + C − 180°. By Girard's theorem the area is Area = E(radians) × R², with E(radians) = E × π / 180. The area comes out in the square of whatever unit you gave the radius: kilometres give km², metres give m². If the radius is left empty it is 1, so the area is then a pure fraction of R². The tool checks that the triangle can exist on a sphere: each angle must be between 0° and 180°, the sum must be above 180° (and so below 540°), and for every pair of angles, their sum minus the third must stay below 180° (170°, 170°, 20° fails this test). The radius must be above zero, and a radius that cannot be read as a number, such as one typed with a decimal comma, shows no result instead of defaulting to 1.

Common mistakes

• Entering angles in radians or grads. The three angles must be in degrees; 1.5708 would be treated as a very small angle and the sum would fail the above-180° test.
• Entering the angles of a flat triangle (such as 60°, 60°, 60°). On a sphere the sum is always more than 180°, and a sum of 180° or less returns no result.
• Entering three angles that add up to more than 180° but cannot form a real spherical triangle, such as 170°, 170°, 20°. The tool shows no result because two large angles cannot be balanced by a tiny third one.
• Entering a diameter instead of the radius. The area scales with the square of R, so a diameter gives an area four times too large.
• Mixing the units of area and radius. The result is in the square of the radius unit, so a radius in km gives km², and the label "R² units" is only a reminder. Type the radius with a decimal point (6371.5), because 6371,5 shows no result.