Math 101 for Pilots

From calculating the reciprocal course to stabilized descents, here's a primer of basic mathematics pilots need to know.

Full disclosure: I never enjoyed math class. Especially story problems. They seemed cruel and unusual punishment. My attitude changed when I became a pilot, learned how to use a mechanical E6B and the performance charts in the pilot operating handbook (POH), and learned to fly instruments using the power-performance concept. These were practical applications of story problems.

Since you can’t or shouldn’t have your head buried in a calculator when you are flying, it behooves you to be able to do some quick math in your head. 

Calculating Reciprocal Course

Let’s start with an easy one: determining the reciprocal course. 

For example, the airplane is flying a heading of 030. You need to fly the reciprocal, and

the only navigational aid you have is the whiskey compass. Add 200 to 030 

(200 + 30 = 230), then subtract 20 because the reciprocal is 180 degrees off the initial heading. You need to fly 210.

Determining Bank Angle for a Standard Rate Turn

I learned this one from a World War II aviator. To calculate the bank angle for a standard rate turn, take the indicated airspeed, divide it by 10, then add five to the value. For example, if the indicated airspeed is 100 knots/10 = 10 + 5 = 15 degrees. Verify this with the attitude indicator and the turn coordinator the next time you fly. 

Time to Climb

Presuming that climbs will be made at 500 fpm, subtract present altitude from target altitude then divide by 500.

For example, the airplane is at a traffic pattern altitude of 1,500 feet msl. The target altitude is 5,500 msl. 5,500 – 1,500 = 4,000. Divide 4,000/500 fpm = 8. It will take approximately eight minutes to get to target altitude.

Descent Rate

Although we’ve never had an airplane stay stuck up there yet, you don’t just pull the throttle back and let gravity take over during a descent. 

Descend to pattern altitude too soon, and you might put yourself in a risky position should there be an uncommanded loss of engine power when you are beyond gliding range to the airport. Delay the descent and you’re behind the airplane, fighting to slow down and get down. Planning is key. You need to determine when to start your descent and the rate of descent.

The Math Behind a Stabilized Descent

Three degrees is the usual angle of descent—when you look at an instrument approach plate and note a descent angle greater than 3 degrees, you can expect an obstacle in the path. 

The time of descent is based on ground speed. In addition, you need to determine the rate of descent.

There are a few ways to determine this:Take your ground speed, divide it in half, and add a zero to the end to determine how many feet per minute you need to get stable descent. 

For example, if your ground speed is 100 knots, divided in half that gives you 50—add another zero, and you get 500 fpm. 

Either refer to the POH or experiment to determine the appropriate power setting and pitch to achieve this. If you pursue an instrument rating, this is a skill you will need because a stabilized constant rate descent will keep you on a  glide slope. For VFR pilots, a stabilized descent keeps you on the PAPI or VASI.

Although there are apps that will do these calculations for you with the push of a button, it’s a good idea to practice them every now and then to make sure you don’t forget how. If the avionics go Tango Uniform or the portable electronic devices overheat in the cockpit—we’ve all been there—you will be happy you remember.


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