A triangle with too many degrees

Every school triangle has angles that add up to 180°. Draw one on a ball and the rule breaks: the angles add up to more, and how much more tells you the triangle's area.

A triangle made of great circles

On a sphere, the straightest possible line is an arc of a great circle: a circle whose centre is the centre of the sphere, like the equator. Pick three points and join each pair by the shorter great-circle arc. That is a spherical triangle. Its angle at a corner is the angle between the two arcs meeting there, measured standing on the surface.

Why it is worth a second look

We live on a near-sphere, so this is the geometry of maps, flight paths and land surveys. Todhunter's 1886 textbook notes that the triangles of the survey of Great Britain were computed with it. It also shows the 180° rule belongs to the flat plane, not to triangles in general.

How much more than 180°

Todhunter's Article 32 proves that the three angles add up to more than two right angles and less than six. In degrees, that is more than 180° and less than 540°. The upper limit holds because each angle is under 180°; the lower one is proved through a partner figure called the polar triangle.

The amount over 180° has a name: the spherical excess. Measure it in radians, where a half-turn is π, and it is A + B + C − π.

The excess is the area

The proof uses slices of orange. A lune is the region between two half great circles that meet at opposite poles. The whole sphere is a lune with an angle of a full turn, 2π, and lunes grow in step with their angle. So a lune of angle A covers A/2π of the surface 4πr², where r is the radius. That is an area of 2Ar² (Article 96).

Now extend the three sides of a triangle all the way round. Each corner sits at the tip of a lune that contains the triangle. The three lunes together cover a hemisphere, with the triangle counted two extra times, once one far-side piece is swapped for its equal-area mirror. Their areas add to 2(A + B + C)r², and that equals the hemisphere 2πr² plus twice the triangle. Rearranged, the area of the triangle is (A + B + C − π)r² (Article 97). The excess must be in radians here. Put degrees into the formula and the answer is wrong by a factor of 180/π, about 57.

A worked example, ours rather than Todhunter's: start at the North Pole, walk down to the equator, turn left through 90°, walk a quarter of the way round, turn left through 90° again and walk home. Every corner is a right angle. The sum is 270°, the excess is 90°, which is π/2 radians, and the area is πr²/2. That is exactly one eighth of 4πr²: eight such triangles tile the sphere.

The same argument extends to any spherical polygon that can be cut into triangles: the area is the angle sum minus (n − 2)π, times r² (Article 99).

Why flat maps usually work

Shrink the triangle and its area shrinks, so the excess shrinks with it. Legendre's theorem (Article 106) makes this exact enough to use. For a triangle whose sides are small compared with the sphere, each angle exceeds the plane-triangle angle with sides the same lengths as the arcs, by one third of the excess. So a surveyor can use flat trigonometry and then correct it.

In short

On a sphere, the angles of a triangle add up to more than 180°. The surplus, in radians, times r² is the triangle's area. Three lunes and a hemisphere prove it. Tiny triangles are nearly flat.