Triangles on a Sphere Break the 180 Rule

Straight lines on a sphere

A triangle has three straight sides. On the curved surface of a sphere nothing can be straight in the flat-paper sense, so the sides are the shortest routes along the surface. The shortest route between two points of a sphere runs along a great circle, a circle drawn on the sphere whose center is the center of the sphere. A great circle cuts the sphere into two equal halves.

On a globe, the equator is a great circle, and so is every meridian, the line of longitude running from the North Pole to the South Pole. The other lines of latitude are not: their centers lie above or below the center of the Earth. A triangle on a sphere has three sides that are arcs of great circles.

A triangle with three right angles

Start at the North Pole and walk straight down a meridian to the equator. Turn and walk along the equator a quarter of the way around the Earth. Turn again and walk straight back up another meridian to the pole. The walk has three straight sides, so it is a triangle.

Every meridian meets the equator at a right angle, so the two corners on the equator are 90° each. The angle at the pole is the angle between the two meridians, and a quarter of the way around the equator is a quarter of a full turn, so that angle is 90° too. The three angles add up to 90° + 90° + 90° = 270°, which is 90° more than 180°.

Down a meridian, a quarter of the way along the equator, and back up another meridian: three corners of 90° each.

Why the flat proof fails

On flat paper the angles of a triangle add up to 180°, and the proof uses a parallel line. Draw the line through the top corner parallel to the base. Alternate angles are equal, so the two base angles appear again at the top corner, one on each side of the top angle, and the three together fill a straight line: 180°.

A sphere has no parallel straight lines. Any two great circles cross, and in fact cross twice: two meridians meet at the North Pole and again at the South Pole. The triangle above shows the difference. On flat paper, two lines that both meet a third line at 90° are parallel, since their co-interior angles add up to 90° + 90° = 180°, and they never meet. The two meridians both meet the equator at 90°, and they still meet at the pole.

So the 180° rule is not a fact about every surface. It follows from parallel lines, and it holds on a flat surface, where they exist.

The bigger the triangle, the bigger the sum

Move the second meridian so that the angle at the pole is $\theta$θ. The two corners on the equator stay at 90°, so the three angles add up to $90^{\circ }+90^{\circ }+\theta =180^{\circ }+\theta$90° + 90° + θ = 180° + θ. The amount over 180° is $\theta$θ itself.

The triangle is a slice of the northern half of the sphere, and $\theta$θ decides how big a slice: at $\theta =90^{\circ }$θ = 90° it is a quarter of that half, and at $\theta =45^{\circ }$θ = 45° it is an eighth. Doubling $\theta$θ doubles both the area of the triangle and the amount over 180°. This is true of every triangle on a sphere, not only these slices: the amount over 180° is proportional to the triangle’s area.

The angle at the pole is 60°, so the angles add up to 90° + 90° + 60° = 240°, which is 60° more than 180°. Drag the corner along the equator to make the pole angle 90°.

Why 180° still works on the ground

A small triangle on a large sphere covers a tiny share of it, so its angles add up to only a tiny amount more than 180°. A triangle marked out on a sports field covers so little of the Earth that the extra is far too small to measure. That is why the 180° rule can be used for any triangle measured on the ground, even though the ground is part of a sphere.

For a triangle the size of a continent, it does not. The pole-and-equator triangle covers an eighth of the Earth’s surface, and its angles add up to 270°.