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Radius of Turn

in Aerodynamics

The radius of turn is directly linked to the ROT, which explained earlier is a function of both bank angle and airspeed. If the bank angle is held constant and the airspeed is increased, the radius of the turn changes (increases). A higher airspeed causes the aircraft to travel through a longer arc due to a greater speed. An aircraft traveling at 120 knots is able to turn a 360° circle in a tighter radius than an aircraft traveling at 240 knots. In order to compensate for the increase in airspeed, the bank angle would need to be increased.


The radius of turn (R) can be computed using a simple formula. The radius of turn is equal to the velocity squared (V2) divided by 11.26 times the tangent of the bank angle.

radius of turn calculation

Using the examples provided in Figures 4-48 through 4-50, the turn radius for each of the two speeds can be computed. Note that if the speed is doubled, the radius is squared. [Figures 4-51 and 4-52]

Figure 4-51. Radius at 120 knots with bank angle of 30°.

Figure 4-51. Radius at 120 knots with bank angle of 30°.

Figure 4-52. Radius at 240 knots.

Figure 4-52. Radius at 240 knots.

Another way to determine the radius of turn is speed in using feet per second (fps), π (3.1415) and the ROT. Using the example on page 4-34 in the upper right column, it was determined that an aircraft with a ROT of 5.25 degrees per second required 68.6 seconds to make a complete circle. An aircraft’s speed (in knots) can be converted to fps by multiplying it by a constant of 1.69. Therefore, an aircraft traveling at 120 knots (TAS) travels at 202.8 fps. Knowing the speed in fps (202.8) multiplied by the time an aircraft takes to complete a circle (68.6 seconds) can determine the size of the circle; 202.8 times 68.6 equals 13,912 feet. Dividing by π yields a diameter of 4,428 feet, which when divided by 2 equals a radius of 2,214 feet [Figure 4-53], a foot within that determined through use of the formula in Figure 4-51.

Figure 4-53. Another formula that can be used for radius.

Figure 4-53. Another formula that can be used for radius.

In Figure 4-54, the pilot enters a canyon and decides to turn 180° to exit. The pilot uses a 30° bank angle in his turn.

Figure 4-54. Two aircraft have flown into a canyon by error. The canyon is 5,000 feet across and has sheer cliffs on both sides. The pilot in the top image is flying at 120 knots. After realizing the error, the pilot banks hard and uses a 30° bank angle to reverse course. This aircraft requires about 4,000 feet to turn 180°, and makes it out of the canyon safely. The pilot in the bottom image is flying at 140 knots and also uses a 30° angle of bank in an attempt to reverse course. The aircraft, although flying just 20 knots faster than the aircraft in the top image, requires over 6,000 feet to reverse course to safety. Unfortunately, the canyon is only 5,000 feet across and the aircraft will hit the canyon wall. The point is that airspeed is the most influential factor in determining how much distance is required to turn. Many pilots have made the error of increasing the steepness of their bank angle when a simple reduction of speed would have been more appropriate.

Figure 4-54. Two aircraft have flown into a canyon by error. The canyon is 5,000 feet across and has sheer cliffs on both sides. The pilot in the top image is flying at 120 knots. After realizing the error, the pilot banks hard and uses a 30° bank angle to reverse course. This aircraft requires about 4,000 feet to turn 180°, and makes it out of the canyon safely. The pilot in the bottom image is flying at 140 knots and also uses a 30° angle of bank in an attempt to reverse course. The aircraft, although flying just 20 knots faster than the aircraft in the top image, requires over 6,000 feet to reverse course to safety. Unfortunately, the canyon is only 5,000 feet across and the aircraft will hit the canyon wall. The point is that airspeed is the most influential factor in determining how much distance is required to turn. Many pilots have made the error of increasing the steepness of their bank angle when a simple reduction of speed would have been more appropriate.

 
1 Aidan Monaghan March 15, 2010 at 10:40 am

Question: Is there a formula or program for calculating the turn radius for descending turns?

Thanks.

2 Karl September 19, 2010 at 4:22 am

Hi! In the example above you have used a constant of 11.26 – I presume this constant is in fact no constant but will vary with bank angle (related to G-loads)? How is this number derived? What will the number be for 25 or 35 degree bank?

Brgds, Karl

3 Paul Beardsell June 20, 2011 at 8:52 am

Very poor and dangerous advice. Of course, if you’re going to fly at the same angle of bank the slower aircraft has a smaller radius. But the faster aircraft can bank more without stalling. Do the maths and you see better to bank more, this requires higher speed, and results in higher G-loads on the airframe. The smallest radius turn occurs where you fly fast enough and at a large enough angle of bank to load the aircraft to its maximum G and where that airspeed coincides with the stall speed at that angle of bank. Believe it or not, that is the truth. To prove the presented article wrong imagine one is flying at the stall speed, straight and level: What is the smallest radius of turn? Infinite! If you turn, you stall. So faster is better in at least this example. The answer remains: For a level (i.e. height maintaining) turn at minimum radius: Fly at maximum bank angle which at a speed which corresponds to max G-load at the stall speed for that bank angle. I.e. bank as much as possible at Va.

4 Tom M. September 9, 2011 at 11:47 am

I wish that you would have used the original formulas, instead of using constants in the formulas like 11.26, etc. To answer Karl’s question, this value is due to the particular units being used in the equation, which forces you to use those units. So V must be in knots, and R will be in feet. If this 11.26 value wasn’t there and the original formula was used, then you could specify the units you wanted, with your own conversion factors. For example:

R = V^2 / (tan(bank angle) * g)

To get R in feet, V would be in ft/sec, g in ft/sec^2. If you want to use other units, you just need them to work out correctly in the end to give you the units of R you are looking for…

I hate seeing mysterious numbers within a formula with no appartant meaning.

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