Why does GPS need four satellites when three spheres seem enough?

Your phone finds its position by measuring how long signals take to arrive from satellites. Each GPS satellite carries an accurate record of its own position and time and broadcasts it continuously. The receiver turns the arrival times into time-of-flight values; those are ranges plus an unknown clock difference times the speed of light, and they are called pseudo-ranges.

That unknown is why you need four satellites, not three. It is sometimes incorrectly said that the user is at the intersection of three spheres. That holds only if the receiver's clock is synchronised with the satellites'. The receiver's clock is usually not of the same quality, so the pseudoranges differ greatly from true distances. A receiver therefore needs at least four satellites to compute four unknowns: three position coordinates and the deviation of its own clock from satellite time.

The system is built to make this work everywhere. Satellites orbit at an altitude of about 20,200 km, making two orbits each sidereal day, and the orbits are arranged so that at least six satellites are always within line of sight from anywhere on Earth. GPS also corrects its time signals for two relativistic effects, from relative velocity and from gravitational potential.

The system began in 1973; a full constellation of 24 satellites became operational in 1993. Civilian use was at first limited to about 100 m accuracy by Selective Availability, a deliberate error that military receivers could correct. President Clinton ordered it disabled on May 1, 2000, and GPS was then accurate to about 5 metres.