In aviation, there are actually many different types of altitudes we use, including indicated, true, pressure, density, and absolute.
Indicated altitude is exactly what it sounds like: it’s what is indicated on the altimeter.
True altitude is your actual height above sea level, and is also known as Mean Sea Level, or MSL. On a standard day (29.92 inches of mercury/ 1013.24 millibars air pressure and 15 degrees Celsius) and with a properly set and calibrated barometric altimeter, your indicated altitude will be the same as your true altitude. Due to weather systems, though, atmospheric pressure is seldom standard. On a non-standard day, an altimeter set to 29.92 will then display your…
Pressure altitude, or your height above the standard datum plane of 29.92 inHg/1013.24 mb. We can get the altimeter to show true altitude again by adjusting it for non-standard pressure. This is done with a small knob on the side. Above certain altitudes where there is no danger of hitting any terrain (18,000 ft in the US), regulations require that all pilots return their altimeters to the 29.92/1013.24 setting. These are called Flight Levels (FL), which you may hear from the captain over the intercom. “Ladies and gentlemen, we are now cruising at flight level 350…” This roughly translates to 35,000 feet, though it will be variable depending on weather systems.
Density altitude is your pressure altitude corrected for non-standard temperature. Basically, this is the altitude that the plane “thinks” it’s flying at, performance-wise. On a hot day, the air is less dense than normal. This means the engine cannot produce as much power, and the wings cannot produce as much lift. Pilots calculate their takeoff, climb, and landing performance by using density altitudes. Performance charts found in aircraft operating manuals automatically adjust for this, so it’s rather easy to do. Density altitude, however, plays a very big role in the decision making process of pilots, particularly those that fly in the mountains or don’t fly jet aircraft. Humidity also can play a big factor, but it is generally not included in aircraft performance charts.
Absolute altitude is your height above ground level, and is also known AGL. This is what a radar altimeter measures.
Radar altimetry in aircraft today is rather different than what you’ve described, though. Now, a lot of aircraft do have that basic radar-based system in them, but they are mostly used as backup devices these days. Primarily, though, they use something called TAWS… Terrain Awareness Warning System. Basically, it’s all computerized. The avionics have a highly precise topographical digital maps built in them. The system will then use this map to tell how close you are to terrain, how long it will be before you fly into terrain, and how fast you’re approaching it. It offers several advantages over basic radar altimetry, most notably that it can see ahead of the aircraft, rather than just straight down. So if you’re flying straight at a cliff face, you’d get an aural warning, whereas with basic radar altimeters, you wouldn’t know until it’s too late. Also, since it’s digital, it can be installed in any aircraft, not just large aircraft. If you’re descending too fast into terrain, it’ll start shouting “SINK RATE, SINK RATE, WHOOP WHOOP PULL UP, PULL UP” at you, which can be very annoying if you’re doing it on purpose to land. It’s also able to automatically change these warnings depending on where you’re at and what phase of flight you’re in. If you’re closer to your destination, the tolerances are a lot lower, so it won’t beep at you as you’re coming in to land. If you’re in cruise, the tolerances are a lot higher. Real radar altimeters are really used only as a backup device these days.
As far as GPS goes, yes, you can use it to derive your altitude, but it’s really inaccurate. Too much so to use in flight. At my job, we use a system that is extremely precise. I’ve seen it triangulate our position with 12 GPS and 4 GLONASS satellites at once, and it’s always accurate within a foot (usually centimeters) laterally. Even so, the GPS derived altitude is usually off by 100+ feet, sometimes as much as 300-400. That may sound pretty good, but when aircraft are being separated by only 500 feet, it’s not nearly accurate enough. The FAA is really trying to pump out GPS WAAS instrument approaches, but they only provide lateral navigation. They can provide vertical navigation, but they use a trigonometry algorithm and the barometric altimeter to figure out what altitude you should be at.