Turn Radius: Formula, Calculation, and Examples

Turn radius is the radius of the circular path followed while turning. Use the calculator to enter speed and bank angle, and it returns the radius in metres; the result assumes a coordinated, level aircraft turn.

Formula

For a coordinated, level turn by an aircraft, the horizontal turn radius is r = v² / (g tan θ). Here, r is radius in metres, v is speed in metres per second, g is gravitational acceleration (9.80665 m/s²), and θ is bank angle in degrees. The calculator also gives the diameter, which is twice the radius, and converts speed to kilometres per hour.

This formula describes the path of the aircraft’s centre of mass in a steady turn. It is not a universal formula for every vehicle or turning situation: cars, ships, and objects following a road or track need different inputs and models. A vehicle turning along a flat circular path can instead be analysed with geometry if its path is already known.

Worked Example

With the defaults, speed is 50 m/s and bank angle is 30°. The calculator uses g = 9.80665 m/s²:

1. Convert the angle’s tangent: tan(30°) ≈ 0.577350.
2. Square the speed: 50² = 2,500 m²/s².
3. Calculate the denominator: 9.80665 × 0.577350 ≈ 5.6617 m/s².
4. Divide: r = 2,500 / 5.6617 ≈ 441.57 m.
5. Diameter = 2 × 441.57 = 883.14 m; speed = 50 × 3.6 = 180 km/h.

What Turn Radius Means

Radius is the distance from the centre of a circular path to the path itself. A smaller radius means a tighter turn; a larger radius means a broader turn. Diameter is the full distance across the circle through its centre, so it is twice the radius. The path length for a complete circle is the circumference, 2πr, but that is not the same as either radius or diameter.

“Turn radius” can refer to different things depending on context: the aircraft’s flight path, a car’s path around a corner, a road’s curve, or a vehicle manufacturer’s turning-circle measurement. Check which path is being measured before comparing numbers. For example, a car’s quoted turning circle may be a diameter and may describe the outer edge of a tyre path, not the radius of the vehicle’s centreline.

How Speed and Bank Angle Affect Radius

The equation shows two key relationships. At a fixed bank angle, radius grows with the square of speed: doubling speed makes the calculated radius four times as large. At a fixed speed, increasing bank angle increases the tangent in the denominator and reduces the radius. These are mathematical relationships within the stated steady, coordinated-turn model, not recommendations for how to fly.

For illustration, at 50 m/s and a 45° bank, tan(45°) is 1, giving a radius of about 255.0 m. At the same speed and 30° bank, the radius is about 441.6 m. In both cases, changing speed has a strong effect because speed is squared.

At a bank angle of zero, the formula has a zero denominator: a level aircraft with no bank does not make a steady horizontal circular turn under this model. The calculator therefore sets the angle’s lower limit slightly above zero. Near zero, the calculated radius can become extremely large. The upper limit also avoids a 90° bank, where the tangent is undefined.

When to Use This Calculation

This calculation is useful for understanding or estimating the geometric radius of a steady, coordinated, level aircraft turn when speed and bank angle are known. The speed should be the speed appropriate to the flight condition being analysed; do not substitute a ground speed without considering wind and the intended reference frame. The formula itself does not account for wind, acceleration during a turn, changing bank, climb or descent, or a slipping or skidding turn.

For a road curve or a car making a turn, use measurements or a model that matches the actual path and constraints. A simple geometry calculation can find radius from a measured diameter (r = d/2), or from a circumference (r = C/(2π)). Those geometric relations do not predict whether a vehicle can safely negotiate the curve. This page’s aircraft formula is not a vehicle-handling or route-planning tool.

Units, Accuracy, and Common Errors

• Convert speed before entering it. The calculator expects m/s. Convert kilometres per hour to m/s by dividing by 3.6; convert miles per hour by multiplying by about 0.44704. For example, 180 km/h is 50 m/s.
• Enter bank angle in degrees. The expression uses a degree-based tangent. A calculator or spreadsheet using radians needs the angle converted to radians first.
• Do not confuse radius and diameter. A result of 441.57 m means a diameter of 883.14 m, not a circle 441.57 m across.
• Keep precision until the end. Rounding speed or the tangent early can noticeably change the result because speed is squared. The displayed values are rounded to two decimal places.

The calculated output is an idealised value, not a measurement or a guaranteed trajectory. Real flight conditions and aircraft performance can differ from the assumptions. The formula also says nothing by itself about operational limits or safe manoeuvres; those require appropriate training, aircraft information, and current conditions.

Frequently Asked Questions

What is the formula for turn radius?
For a coordinated, level aircraft turn, use r = v²/(g tan θ), with speed in m/s, gravitational acceleration in m/s², and bank angle θ in degrees when using a degree-based tangent.

How do I calculate turn radius from diameter?
Divide the diameter by two: r = d/2. Make sure the measurement refers to the same circular path whose radius you need.

How does speed affect turn radius?
At a fixed bank angle in the aircraft formula, radius is proportional to speed squared. Doubling speed produces four times the radius, assuming the same idealised conditions.

Can I use this formula for a car?
Not as a general car-turning calculator. The displayed formula models a banked aircraft turn. A car’s turning path depends on its geometry and steering, among other factors; use an appropriate vehicle-specific method.

Why does a small bank angle give a large radius?
A small bank has a small tangent, so the denominator is small and the calculated radius grows. At zero bank, the formula cannot give a finite steady horizontal turning radius.

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in v | Speed | m/s | 50 | 0
in b | Bank angle | degrees | 30 | 0 | 89.9
out r | Turn radius | m | v ^ 2 / (9.80665 * tand(b)) | 2
out d | Turn diameter | m | 2 * r | 2
out s | Speed | km/h | v * 3.6 | 2