Air has weight, and the weight of the air above you is what a barometer measures. At sea level, that weight presses down with about 1013 hectopascals. As you climb, less air sits above you, so pressure falls. The fall is steady but not linear: pressure drops fast near the ground, about 1 hectopascal every 8 metres, and more slowly higher up, because the air itself gets thinner. This relationship lets you turn a pressure reading into an altitude estimate, and back again.
The maths behind this tool is the standard atmosphere model. It assumes a sea-level pressure of 1013.25 hectopascals, a sea-level temperature of 15 degrees, and a steady temperature fall with height up to 11 kilometres. The formula is pressure equals 1013.25 times one minus a small constant times height, raised to the power 5.25588. The inverse formula runs the other way, turning any pressure back into a height. Both directions live in this tool.
Here is a worked example. At 1000 metres, the formula gives about 898.7 hectopascals, roughly 114 hectopascals below sea level. At 2000 metres it gives about 794.9 hectopascals. Going the other way, a barometer reading of 900 hectopascals corresponds to about 988 metres in the standard atmosphere. A hill walker with a barometer watch uses exactly this maths, and so does an aircraft altimeter, which is really a barometer with its dial marked in feet instead of pressure.
Aircraft use this relationship every day, with one vital correction. Pilots set their altimeters to the local sea-level pressure, called QNH, reported by air traffic control. Without that correction, a weather change could make the altimeter read hundreds of feet wrong, which is dangerous near mountains. When every aircraft in an area uses the same QNH, their altitude readings agree, even if the standard atmosphere does not match the real weather that day.
Weather itself is the biggest source of error. A deep low-pressure system can drop sea-level pressure to 980 hectopascals or below, while a strong high can push it above 1040. If you feed a storm-day pressure into the standard formula without correcting for sea-level pressure, your altitude estimate can be wrong by several hundred metres. The tool assumes standard sea-level pressure, so on unusual weather days, treat the altitude as a rough guide, not a measurement.
Pressure also changes with temperature, which the simple model partly handles. Cold air is denser and packs more tightly, so pressure falls faster with height on a cold day than on a warm day. The standard model uses an average temperature profile, so it is most accurate on a mild day and least accurate in extreme cold or heat. High-altitude mountaineers know this well: in deep cold, true altitude can be lower than the altimeter shows, a trap called cold temperature error.
Beyond aviation, this relationship shows up in cooking, sport, and science. Water boils at a lower temperature at altitude because the pressure is lower: about 97 degrees at 1000 metres and about 93 degrees at 2000 metres, so pasta and rice take longer. Endurance athletes train at altitude because thinner air carries less oxygen. Weather stations report sea-level pressure, not station pressure, precisely so that readings from a mountain town and a coastal city can be compared fairly.
To use this tool well, enter exactly one of the two values and leave the other blank. If you know your altitude from a map and want the expected pressure, enter the altitude. If you have a barometer reading and want an altitude estimate, enter the pressure, and remember the weather caveat. For the best altitude from pressure, first correct your reading to sea level if your barometer offers that option, or compare against a known nearby altitude on the same day.