Rate of Turn Calculator

Bank angle

True airspeed

Rate of turn

Enter the bank angle and the true airspeed to get the rate of turn, or fill any two of the three fields and leave the unknown blank. A standard-rate turn is 3 degrees per second, a full circle in two minutes.

Show turn geometry

Turn the corner into a full circle and add the turn radius, the diameter, and the distance flown in one 360-degree lap.

Add the stall speed check

Enter your level-flight stall speed to see how much the bank angle raises the stall speed in the turn, the accelerated stall.

Add heading-change timing

Enter a heading change, such as a 180-degree procedure turn, to get the time it takes at this turn rate and at the standard rate.

Please input values into the fields

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Show the rate-vs-bank chart

Plot how the turn rate climbs with bank angle for your airspeed, against the standard-rate reference, with a matching bank-angle table.

Show the standard-rate benchmark

Compare the bank angle needed for a standard-rate turn across a trainer, a GA aircraft, a turboprop, a jet, and an airliner, with your airspeed added.

These bank angles assume the airspeed is the true airspeed. At high altitude true airspeed is well above the indicated airspeed, which is one reason jets bank more steeply for the same rate than the numbers on the airspeed indicator suggest.

The rate at which the nose of an aircraft turns in a coordinated turn ultimately depends on two factors, the angle of bank and airspeed. This calculator computes the turn rate from these two values. It can also be used in reverse to determine either the required angle of bank or airspeed for a given turn rate. Enter any two of the three values and leave the third field blank.

What is the speed of the curve?

The rate of turn is the speed at which an aircraft's nose turns during a coordinated turn, typically expressed in degrees per second. The most commonly used value is standard rate, 3 degrees per second. In this case, the aircraft completes a full circle of 360 degrees in exactly two minutes. Half standard rate, or 1.5 degrees per second, is referred to as half-standard and is common on faster-moving aircraft.

The rate of turn depends on two factors: the angle of bank and airspeed. The greater the angle of bank, the more lift is directed horizontally, causing the nose to turn faster. The higher the airspeed, the larger the radius of turn, which causes the nose to turn slower. On approach and holding patterns, pilots and controllers rely on standard rates of turn because predictable turns make it easier to predict aircraft spacing.

Formula for curve speed

This turn rate is directly derived from the circular motion. In a coordinated bank turn, the horizontal component of lift produces a centripetal force that curves the flight path. The turn rate can be calculated as follows:

\omega = \frac{g \, \tan(\theta)}{V}

where ω is the rate of roll, g is the acceleration due to gravity (9.80665 meters per second squared), θ is the angle of attack and V is the true airspeed. As the airspeed is in the denominator, an aircraft will roll more slowly at a given angle of attack as it flies faster. Also, because of the tangent function used to calculate the angle of attack, the rate of roll increases dramatically as the angle of attack approaches 90 degrees.

Example calculation:

Let's take a light aircraft with a speed of 120 knots. The speed is about 61.7 meters per second and the plane has an angle of attack of 20 degrees.

\omega = \frac{9.80665 \times \tan(20^\circ)}{61.7} = \frac{9.80665 \times 0.364}{61.7} \approx 0.0578\ \text{rad/s}

To convert radians per second to degrees per second, multiply the value by π. The result is approximately 3.3 degrees per second, which happens to be exactly the standard rate of precession. Thus, airplanes generally need a pitch angle of about 20 degrees in order to achieve the standard rate of precession at typical cruise speeds.

Expressed as a load multiplier:

The same rate of roll can also be expressed using a load factor instead of the angle of bank. The load factor n indicates how much larger the load is that the wings of an aircraft have to bear during a turn, and it equals the reciprocal of the cosine of the angle of bank. Since there is a relationship between the tangent function and the load factor, the rate of roll can also be expressed as:

\omega = \frac{g \, \sqrt{n^2 - 1}}{V}

The results that come from the two formulas are identical. The formula using load factor is more practical because the load factor represents what the airplane and pilot actually experience. At a 60 degree angle of attack, the load factor is 2, which doubles the weight of everything in the airplane and increases the stall speed. This calculator shows both the load factor and rate of turn for that reason.

Reverse calculation:

This relationship contains three variables but only one equation, so the equation can be rearranged to solve for any of the missing values.

\theta = \arctan\!\left(\frac{\omega V}{g}\right) \qquad V = \frac{g \, \tan(\theta)}{\omega}

It is possible to determine both the angle of attack that will produce a standard rate turn at a given airspeed and also calculate the airspeed corresponding to a particular angle of attack and rate of turn. No model selection is required. Leave one field blank to tell the calculator which value to compute.

This table represents the same relationship in three different ways, corresponding to each of the three possible missing quantities.

To find	Formula	In words
Rate of turn	omega = g tan(theta) / V	Gravity times tangent of bank, over airspeed
Bank angle	theta = arctan(omega V / g)	Arctangent of rate times airspeed over gravity
Airspeed	V = g tan(theta) / omega	Gravity times tangent of bank, over rate

Standard curve rate and two-minute curve.

At standard rate of turn the nose of the airplane is changing direction at 3 degrees per second. It takes two minutes to make a 360 degree turn and one minute for a 180-degree turn. Instrument flight procedures are based on this value, and there is a gyroscopic compass in the cockpit with markings corresponding to it. When performing a standard rate turn, the wing should be aligned with these markings.

A quick way to determine the required angle of attack for standard rate is to divide the airspeed in knots by 10 and add 7. At an airspeed of 120 knots, this would be 12 + 7 or approximately 19 degrees, a value close to that given by this calculator. The faster the aircraft flies, the greater the required angle of attack for standard rate. Many jets use half standard rate when they exceed about 250 knots in order to maintain comfortable angles of attack.

Circle radius and one percent rule

Turn rate and turn radius are two different ways of describing the same turn. Turn rate is how quickly the nose of the aircraft changes direction while turn radius is the size of the circle being made. The two values are inversely proportional, with the turn radius equal to airspeed divided by turn rate.

r = \frac{V}{\omega}

This can be used to derive a simple rule of thumb for standard curve speed. The diameter (in nautical miles) is approximately the groundspeed (in knots) divided by 100. At a groundspeed of 100 knots, the diameter would be about one nautical mile, which equates to a radius of half a nautical mile. When you open up the curve geometry pane, it will show you the radius, diameter and distance traveled during a complete revolution of the circle based on your own values.

Load coefficient, angle of attack, flow separation

The load factor is a hidden component of banked turns. At 30 degrees the load factor is 1.15, at 45 degrees it's 1.41 and at 60 degrees it's 2. Since stall speed increases in proportion to the square root of the load factor, the stall speed increases by about 41 percent at a bank angle of 60 degrees. This is an example of acceleration induced stall. Sudden turns can cause many accidents especially at low altitudes and speeds when transitioning from base to final approach.

If you open the Stall Speeds section and enter your stall speed at level flight, then the calculator will show how much higher your stall speed becomes with the angle of attack that you have selected. This is a quick way to check realistic conditions before flying steep turns at low altitude and speeds.

Application of curve speed

Pilots who fly instrument flight rules maintain a standard rate of turn during approach, hover and waypoints. This makes it easier to predict timing. Air traffic controllers also take the standard rate of turn into account when planning aircraft for approach in order to set spacing.

Flight planning software and simulators use the same formula to predict where an aircraft will be during a turn. Autopilots and flight director systems command the angle of attack so that the desired turning rate is achieved. Outside the cockpit, this relationship serves as a classic example in physics and aerodynamics courses dealing with coordinated turns.

Assumptions and Limitations

This formula assumes that the aircraft is in a coordinated turn. This means that the ball in the slip indicator is centered and there is no sideslip or skid. Also, this formula uses true airspeed, not indicated airspeed. At high altitudes, true airspeed is much higher than indicated airspeed, so the actual rate of turn will be lower than the value derived from indicated airspeed.

The rate of turn indicates the path that an aircraft traces in the air. Since wind changes the path of the airplane relative to the ground, ground tracks and radii will differ from those without wind. Gravity also decreases slightly with altitude but below 40,000 feet this difference has less than a half percent effect on rate of turn.

Frequently asked questions

What is a standard curve rate?

At a standard rate turn the nose of the airplane changes direction at three degrees per second so that a 360 degree turn is completed in exactly two minutes and an 180 degree turn in one minute. This is the standard rate turn used in most instrument flight procedures. The turn coordinator has markings for this, and when performing a standard rate turn the wings are brought into position to match these marks. High speed airplanes usually fly at half standard rate of 1.5 degrees per second to maintain proper angle of attack.

What is the formula for rate of curvature?

The formula for rotational speed is ω = g*tan(θ) / V where g is the Earth's gravitational acceleration of 9.80665 m/s², θ is the angle of attack and V is the relative airspeed. The result will be in radians per second; to convert it into degrees per second multiply by 180/π. Rearranging the formula either gives the angle of attack θ = arctan(ω*V/g), or calculate the relative airspeed, V=g*tan(θ)/ω.

What is the difference between speed and radius?

The rate of turn measures how fast the nose of the airplane turns and is expressed in degrees per second. The turning radius indicates the size of the circular path and is usually measured in feet, meters or nautical miles. Both values are inversely proportional to each other with relative airspeed playing a role. This means that the radius is equal to the relative airspeed divided by the rate of turn. The higher the rate of turn, the tighter and smaller the turn will be. At standard rates of turn, the diameter (in nautical miles) is approximately the ratio of ground speed (in knots) over 100.

Why do planes need a higher angle of attack at faster speeds for the same turn rate?

At higher relative speeds the aircraft covers more distance per second. So to change direction at the same angular rate of turn, the flight path must be deflected through a greater angle. This requires more centripetal force, which causes the lift vector to deviate further from vertical and a larger angle of attack is required. A training aeroplane flying at 90 knots will achieve standard rate turn at a small angle of attack, whereas a jet aircraft flying at 250 knots needs about 32 degrees.

Does the angle of attack affect the flow separation speed in a turn?

Yes. The load factor increases with bank angle, and the stall speed increases proportionally to the square root of the load factor. At a 45-degree bank, the stall speed will increase by about 19 percent, and at a 60-degree bank it will increase by about 41 percent. This accelerated stall is actually dangerous in steep slow turns. An example would be a tight turn when transitioning from cruise to approach. The calculator's stall speed display shows this effect based on the entered stall speed.

Does the speed of the curve change with altitude?

This formula uses true airspeed and gravity. While the effect of gravity is slightly less at higher altitudes, below 40,000 feet this effect on turn rate is less than half a percent and so usually ignored. A larger effect is that at high altitudes true airspeed is significantly greater than indicated airspeed. So for a given bank angle, actual turn rate will be lower at higher altitudes than the value calculated using indicated airspeed.

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Disclaimer: This calculator is provided for general informational and educational purposes only. Our calculators are under active development, and results may be inaccurate, incomplete, or unsuitable for your situation. Always verify the figures independently and seek advice from a qualified professional before relying on them. We make no warranties and accept no liability for any loss or decision arising from use of this tool.

References

Wikipedia: Standard rate turn
Definition of the standard-rate (rate-one) turn, the two-minute turn, and the bank-angle rule of thumb.
Wikipedia: Banked turn
The physics of a coordinated banked turn, including the turn-rate and turn-radius relations from the lift vector.
FAA Airplane Flying Handbook
Reference for coordinated turns, load factor, and the accelerated stall in a banked turn.