How to use the spherical triangle solver
- Choose what you know: three sides, three angles, or a mix such as two sides and the angle between.
- Enter the three values in degrees. Sides are arcs measured as angles at the center of the sphere.
- Optionally enter the sphere radius to get arc lengths and the area.
- Read all six parts and the spherical excess. For side, side, angle (and two angles and a side) both possible triangles are shown.
Formula
Law of cosines for sides: cos a = cos b cos c + sin b sin c cos A
Law of cosines for angles: cos A = -cos B cos C + sin B sin C cos a
Law of sines: sin a / sin A = sin b / sin B = sin c / sin C
Spherical excess E = A + B + C - 180 degrees; area on a sphere of radius R = E (in radians) x R^2
Cases with angles are solved on the polar triangle, in which each side is 180 degrees minus the opposite angle.
Three sides, and an ambiguous case
With sides 60, 70 and 80 degrees the angles are A = 61.5677, B = 72.5892 and C = 89.8143 degrees, so the excess is 43.9712 degrees. For two sides of 50 and 65 degrees with 40 degrees between them, the third side is 36.5346 degrees, the other angles are 55.8061 and 101.8774 degrees, and on Earth (radius 6371 km) the area is about 12,527,370 square kilometers.
With sides a = 60 and b = 70 and the angle A = 50 degrees opposite a, there are two triangles: one with c = 16.5610, B = 123.7771 and C = 14.6035 degrees, and one with c = 104.3987, B = 56.2229 and C = 121.0445 degrees. Three angles of 100, 110 and 120 degrees give sides of 90.1857, 107.4108 and 118.4323 degrees and an excess of 150 degrees.
How spherical triangles differ
On a sphere the three sides of a triangle lie along great circles, and the angles of the triangle always add up to more than 180 degrees. The extra, called the spherical excess, is proportional to the area. This is also why the three angles alone fix the triangle: two spherical triangles with equal angles are the same size, unlike flat triangles.
Navigators and astronomers use these triangles to find distances and directions across the globe. For a flat triangle, use the triangle calculator, and for distances between two places use the great circle distance calculator.
Ambiguous and impossible cases
When you know two sides and an angle not between them, or two angles and a side not between them, two different triangles may fit, one fit, or none. The page lists every solution it finds. It reports an error when the values cannot form a triangle: sides must be under 180 degrees, each side must be less than the sum of the other two, and the three sides together must stay under 360 degrees.
For the angles, the sum must be between 180 and 540 degrees. For the trigonometry behind the scenes, see the trigonometry calculator, and for the surface of a sphere cut by planes, see the spherical cap and zone calculator.
Assumptions and limits
- The sphere is perfect, so for Earth the results are approximations within about half a percent.
- Sides and angles are each between 0 and 180 degrees, and the triangle is the smaller one with all parts under 180 degrees.
- Values are given and shown in degrees with four decimals.
- Arc lengths and area use the radius you enter, in whatever unit it is.
Frequently asked questions
What is a spherical triangle?
A triangle drawn on a sphere with sides along great circle arcs. Its sides are measured as angles at the center.
Why do the angles add up to more than 180 degrees?
Because the surface curves. The extra amount, the spherical excess, grows with the area of the triangle.
Why are there sometimes two answers?
With a side, its opposite angle and one more side (or the dual case with angles), the sine rule can give two valid triangles, just like in flat geometry.
How is the area found?
The area is the spherical excess in radians times the radius squared. On the unit sphere it is just the excess in radians.