Worked examples

Check the geometry before interpreting the kilometres.

Each example uses an explicitly chosen radius. Approximate values below are rounded for reading. You can enter every coordinate as a plain decimal; no device location or external lookup is used.

1. A quarter of a great circle

Enter A = (0°, 0°), B = (0°, 90°) and radius = 6371 km. The points are 90° apart around the equator, so θ = π/2 radians. Multiplying gives 6371 × π/2 ≈ 10007.543398 km. With radius 1 km, the same angle would give about 1.57079632679 km.

2. Crossing the longitude label boundary

Enter A = (0°, 179.5°), B = (0°, −179.5°) and radius = 6371 km. Although the raw longitude labels differ by −359°, their normalized difference is +1°. The shorter arc is 1°, or π/180 radians. The distance is about 111.194926645 km. This is a geometric crossing of the dateline convention, not a travel route.

3. Two labels for the same meridian

Enter A = (45°, −180°), B = (45°, 180°) and radius = 6371 km. The longitudes are aliases, so the normalized longitude difference is zero and the identical latitudes identify the same point. The central angle and distance are both 0.

4. Longitude at the pole

Enter A = (90°, 10°), B = (90°, −170°) and radius = 6371 km. Both entries are the north pole. Longitude has no effect there, so the result is 0 km. Change latitude B to −90° and the points become opposite poles: θ = π and distance ≈ 20015.086796 km. Several shortest arcs connect antipodes.

5. A tiny positive separation stays positive

Enter both latitudes as 89.999999999°, longitude A = 0°, longitude B = 0.000000001°, and radius = 6371 km. The points are extremely close to the north pole but are not that pole, and their longitudes differ. The central angle is about 3.04617419787 × 10⁻²² radians, giving about 1.94071758146 × 10⁻¹⁸ km. The scientific notation preserves a positive numerical result; it does not suggest that these positions are physically measurable to that precision.

6. Radius changes scale, not angle

Enter A = (−20°, 145°), B = (35°, −75°) and radius = 6000 km. The angle is about 2.47484676797 radians, or 141.798274746°. Distance is about 14849.0806078 km. At radius 3000 km, it halves to about 7424.54030390 km while the angle stays unchanged.

7. Invalid decimal bounds are not rounded into range

Latitude 90.000000001° is above the accepted bound, even though a short display might round it to 90. Radius 1000000000.000001 km is likewise too large. Both are rejected. A value such as 1e-9 is not an accepted coordinate spelling: write 0.000000001 instead. Invalid entries leave no stale result available for copying.