Formula & Calculator
Radius of Turn (Coordinated)
Turning radius for a coordinated, level turn as a function of speed and bank angle.
Interpretation
Radius of turn: R = V²/(g·tan(φ)), where V is airspeed, φ is bank angle. It gives the turn radius for a coordinated turn. Example: V=100 m/s, φ=30° → R = 10000/(9.81×0.577) ≈ 1768 m.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| R | Turn radius | m |
| V | True airspeed | m/s |
| g | Gravitational acceleration | m/s2 |
| φ | Bank angle | deg |
What it means
The turn radius in a coordinated, level turn depends on the airspeed and bank angle. This formula comes from the balance of forces: the horizontal component of lift provides centripetal acceleration. A higher bank angle reduces the radius but increases load factor. Turn radius is critical for aircraft manoeuvrability, air traffic control spacing, and dogfighting. It also affects the required runway length for turning onto a taxiway. The formula is used in flight dynamics to evaluate turning performance and to design autopilot algorithms. Understanding this relationship is essential for pilots to plan turns within airspace constraints and for engineers to assess agility.
Worked example
Radius of Turn – Two Examples
Real‑World| Parameter | Value |
|---|---|
| V | 100 m/s |
| φ | 30° |
| Parameter | Value |
|---|---|
| V | 200 m/s |
| φ | 60° |
Common mistakes
- Radius of turn R: R = V² / (g·tan(φ)) – for coordinated turn.
- Bank angle φ: Angle between wings and horizontal.
- Speed V: True airspeed (m/s).
- g: 9.81 m/s².
- Load factor: n = 1/cos(φ) – affects stall speed.
Applications
The radius of turn in a coordinated turn is R = V²/(g·tanφ), where φ is the bank angle. This formula relates turning radius to speed and bank angle. It is used in flight mechanics to design aircraft for manoeuvrability, to assess turning performance in combat, and to plan approach paths. Engineers use it to compute load factors, to size control surfaces, and to ensure that the aircraft can perform required turns within airspace constraints. The radius of turn also influences terrain‑following and obstacle avoidance. By mastering this formula, aerospace engineers can optimise aircraft for mission‑specific turning requirements, balancing agility with structural loads.
- Manoeuvring performance analysis for fighters and trainers
- Design of turn radius and bank angle limits
- Approach path and airspace design (e.g., holding patterns)
- Load factor and structural stress during turns
- Control system requirements for coordinated turns
Frequently Asked Questions
It calculates the turning radius for a coordinated, level turn as a function of speed and bank angle. It is used in aircraft performance, air traffic management, and dogfighting.
V = true airspeed (m/s)
g = acceleration due to gravity (9.81 m/s²)
φ = bank angle (radians)
For a given speed, the turn radius decreases as bank angle increases (since tan(φ) increases). However, the load factor also increases, limiting the maximum bank angle.
Turn radius is proportional to V². Doubling the speed quadruples the turn radius for the same bank angle.
A coordinated turn is one where the aircraft is banked and the yaw rate is matched so that the ball in the turn‑and‑slip indicator remains centred. This eliminates sideslip.
It assumes a level turn (no altitude change), constant speed, and that the lift vector is perpendicular to the wings. It does not account for thrust or drag effects.
- Using degrees directly in tan() without converting to radians in code.
- Ignoring load factor limits (e.g., structural limit).
- Using indicated airspeed instead of true airspeed.
At V = 100 m/s and φ = 45° (0.785 rad), R = 100²/(9.81×tan(0.785)) = 10000/(9.81×1) = 1019 m.
The rate of turn (ω) = V/R = g·tan(φ)/V. For a given bank angle, the turn rate decreases with speed.
Since n = 1/cos(φ), the maximum bank angle is φmax = arccos(1/nmax). For nmax = 3 (typical transport), φmax ≈ 70.5°.