Explain how a phone or gps device works out its position from satellite signals. Show how one satellite gives a sphere of possible places, two narrow it to a circle, three to a point — and why a clock error of one microsecond puts you 300 metres away.

When you open your maps app, your phone already knows where you are. Even in fog, rain, or snow, that blue dot is right. How does it pull that off? The answer is hidden in the sky.

Here’s the road ahead. Your phone measures distance to satellites, and each distance shapes where you could be. One satellite puts you on a sphere. Two narrow that sphere to a circle.

Three pin you to a single point. But a tiny clock error — one microsecond — shifts your answer by 300 metres. Follow each shape, and you’ll see exactly how GPS works. Think of a weather front.

One report tells you it is somewhere in a whole region. A second report narrows it to a line. A third pins it to a town. Your phone does the same with satellites.

One satellite signal says: you are somewhere on a sphere around it. Add a second satellite: the two spheres meet in a circle, so you are somewhere on that circle. A third satellite cuts that circle at one point — your position. But there is a catch.

A phone's clock is slightly off. Light travels about three hundred metres in one microsecond. So a one-microsecond clock error shifts every measured distance by three hundred metres — and your calculated position lands three hundred metres away. Your phone finds your position by timing signals from three satellites: three distances meet at one point.

Let's work through a concrete example. Satellite A sends a signal, and your phone measures 20,000 kilometres to it. One distance alone doesn't tell you where you are — you could be anywhere on a sphere around that satellite. Add satellite B, with a distance of 21,000 kilometres.

The two spheres overlap in a circle. Add satellite C, 19,000 kilometres away. On Earth, that circle is cut down to one point: your position. Now the clock catch.

If the phone's clock is off by one microsecond, every distance is wrong by 300 metres. Light travels 300 metres in one microsecond, so the point moves 300 metres from where you really are — enough to miss a street. So here's the map: one satellite puts you on a sphere, two narrow it to a circle, three fix a point. But a clock error of one microsecond moves the answer 300 metres.

That's why the receiver solves for the clock error too — four satellites finally tell you exactly where you are.