Bank Angle for Coordinated Turn
The bank angle needed for a level coordinated turn: φ = arctan( V² / (g · R) ). Solve for the bank angle, the true airspeed, or the turn radius, and read the aerodynamic load factor and turn rate.
Inputs
Choose what to solve for, then fill in the other two. The calculator uses standard gravity g = 9.80665 m/s². Speed and radius can be in any supported units — they convert automatically. The inverse tangent returns an angle; this calculator displays the bank angle in degrees.
Results
Enter values and press Calculate. Your answer will appear here.
Forces in the banked turn
The aircraft is drawn as seen from directly behind. Lift L acts perpendicular to the wings; its vertical component L·cos φ balances weight W = m·g, and its horizontal component L·sin φ provides the centripetal force m·V²/R that curves the flight path. In a coordinated turn there is no sideways (slip or skid) aerodynamic force — the ball in the turn coordinator stays centred.
Turn geometry from above
Top-down view of the turn. The turn radius R is the distance from the aircraft to the centre of the turn, measured in the horizontal plane. The full circle is one 360° turn; the marker shows your aircraft's position and heading. The turn rate ω = V/R determines how long one full circle takes: t360 = 2π·R / V.
Bank angle vs airspeed
Bank angle rises steeply with airspeed at a fixed radius. Each dashed reference line is a different turn radius; the solid teal line is your radius. The amber dot marks your operating point. The horizontal bands show the corresponding aerodynamic load factor at each bank angle.
What is being calculated
Common examples
| Case | V | R | Bank φ | Load n |
|---|
Computed by the same engine. Illustrative values, not aircraft-specific.
Bank angle and load factor
| Bank angle φ | Load factor n = 1 / cos φ | Stall-speed multiplier √n |
|---|---|---|
| 0° | 1.00 G | 1.00× |
| 15° | 1.04 G | 1.02× |
| 30° | 1.15 G | 1.07× |
| 45° | 1.41 G | 1.19× |
| 60° | 2.00 G | 1.41× |
| 70° | 2.92 G | 1.71× |
| 75° | 3.86 G | 1.97× |
| 80° | 5.76 G | 2.40× |
The load factor shown is the aerodynamic result of a level coordinated turn: n = 1 / cos φ. It should not be interpreted as an aircraft certification limit. Permitted bank angles and manoeuvres depend on the aircraft's approved operating limitations and flight manual.
Key relations and unit conversions
| Quantity | Relation |
|---|---|
| Bank angle for a coordinated turn | φ = arctan( V² / (g·R) ) |
| Turn radius from bank angle | R = V² / (g · tan φ) |
| Airspeed from bank angle and radius | V = √( g · R · tan φ ) |
| Load factor | n = 1 / cos φ = √(1 + (V²/(gR))²) |
| Turn rate | ω = V / R = g · tan φ / V (rad/s) |
| Turn rate in degrees per second | ω° = 57.2958 · V / R |
| Standard rate turn | ω = 3°/s → 360° in 120 s |
| Half-standard rate turn | ω = 1.5°/s → 360° in 240 s |
| Time for one full turn | t360 = 2π · R / V = 2π / ω |
| Centripetal acceleration | ac = V² / R = g · tan φ |
| Stall speed multiplier | Vs,turn / Vs,1g = √n = 1/√(cos φ) |
| Vertical lift component | L · cos φ = m · g |
| Horizontal lift component | L · sin φ = m · V² / R |
| Total lift (per unit mass) | L / m = n · g = g / cos φ |
| Speed of sound (ISA sea level) | a = 340.29 m/s |
| Speed unit conversions | 1 kt = 0.514444 m/s · 1 km/h = 0.277778 m/s · 1 mph = 0.44704 m/s · 1 ft/s = 0.3048 m/s |
| Length conversions | 1 km = 1000 m · 1 nm = 1852 m · 1 mi = 1609.344 m · 1 ft = 0.3048 m |
| Gravity | g = 9.80665 m/s² = 32.174 ft/s² |
Formula
- φ: bank angle from the horizontal, in degrees or radians
- V: true airspeed (m/s in the SI form; the calculator converts from kt, km/h, mph or ft/s)
- g: gravitational acceleration, 9.80665 m/s²
- R: horizontal turn radius (m in the SI form)
Example — a light aircraft in a standard rate turn at 120 kt. V = 120 kt = 61.73 m/s. A standard rate turn is 3°/s, which requires R = V/ω = 61.73 / 0.05236 = 1179 m. Then φ = arctan(61.73² / (9.80665 × 1179)) = arctan(3810 / 11 562) = arctan(0.3296) = 18.25°, with load factor n = 1/cos(18.25°) ≈ 1.052.
In a level coordinated turn, three forces act on the aircraft. Weight W = m·g pulls straight down. Lift L acts perpendicular to the wings. Because the wings are banked by angle φ, the lift vector is tilted; its vertical component L·cos φ must equal the weight to keep the aircraft level, and its horizontal component L·sin φ provides the centripetal force m·V²/R that curves the flight path into a circle. Solving the two equations gives tan φ = V²/(gR), which is the formula this calculator implements.
The load factor n = L/W = 1/cos φ is the ratio of lift to weight. At 30° bank it is 1.15; at 45° it is 1.41; at 60° it is 2.00; at 75° it is 3.86. As load factor increases, stall speed also increases: Vs,turn = Vs,1g · √n. At 60° bank the stall speed is √2 ≈ 1.41 times the 1 g stall speed, and the margin to the stall decreases quickly at steeper angles. The load factor is the aerodynamic result of the manoeuvre, not a certification-category limit — permitted bank angles and manoeuvres depend on the aircraft's approved operating limitations and flight manual.
The turn rate ω = V/R = g·tan φ / V is the angular velocity of the flight path. Standard rate — 3° per second — is the reference used in instrument flying for terminal-area procedures; it completes a full 360° turn in 2 minutes. At higher true airspeeds, maintaining a 3°/s turn rate requires greater bank. Reduced turn rates may be used where specified by applicable procedures or operating rules. The radius in a standard rate turn grows linearly with true airspeed: 120 kt gives about 0.64 nm, 240 kt gives about 1.3 nm, 480 kt gives about 2.5 nm. Holding patterns are sized around these numbers.
The formula assumes the turn is level (constant altitude), coordinated (no slip or skid), and flown at constant airspeed. It ignores the horizontal tail download, the effect of pitch attitude on lift, and the small induced-drag penalty of the increased angle of attack. For an aircraft holding altitude in a steady turn, these are negligible. In a steady climbing or descending helical turn with flight-path angle γ, the load factor is n = cos γ / cos φ, slightly below the level-turn value. Manoeuvres such as a chandelle are not steady turns, so the formula does not apply to them. The formula also assumes no wind: in a moving air mass the ground track is not circular, because the wind shifts the aircraft relative to the ground, but the aerodynamic turn relationship itself is based on the aircraft's motion through the air mass — the bank angle for a given airspeed and air-mass turn radius is unchanged.
Advanced mode adds the total lift required per unit mass, the vertical and horizontal lift components, the centripetal acceleration, the time for one full 360° turn, the stall-speed multiplier, the bank angle required for a standard rate turn at the current airspeed, and the radius of a standard rate turn at the current airspeed.
References
- Wikipedia, Banked turn: derivation of tan φ = V²/(gR)
- Wikipedia, Load factor (aeronautics): n = 1/cos φ and stall-speed effects
- Wikipedia, Standard rate turn: 3°/s and half-standard rate
- FAA Pilot's Handbook of Aeronautical Knowledge: chapters on turns and load factor
- FAA Instrument Procedures Handbook: standard rate turns and holding patterns
- NIST SP 811, Guide for the Use of the SI: unit conversion factors
How to Use This Tool
- Choose what to solve for: bank angle φ, airspeed V, or turn radius R.
- Pick a preset, or type the two known values. Speeds and radii convert between m/s, kt, km/h, mph, ft/s, m, km, ft, nm and mi.
- In Advanced, read the load factor, turn rate, one-turn time, stall-speed multiplier, and the bank angle for a standard rate turn.
- Click Calculate.
- Read the highlighted answer, then check the force diagram, the turn geometry and the bank-vs-speed chart.
What This Tool Offers
- Solves φ = arctan(V²/(g·R)) for φ, V or R
- Automatic unit conversion for speed and length — metric and aviation units
- Aerodynamic load factor n = 1/cos φ with stall-speed penalty √n
- Turn rate ω and time for one full 360° turn
- Comparison against the standard rate turn (3°/s) and half-standard rate
- From-behind force diagram and top-down turn geometry diagram
- Bank-vs-airspeed chart with reference radii and load-factor bands
- Reference table of bank angle vs load factor and unit conversions
Disclaimer. The calculated load factor applies to a level coordinated turn. It should not be interpreted as an aircraft certification limit. Always follow the aircraft's approved operating limitations and flight manual.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| φ | Bank angle | deg |
| V | True airspeed | m/s |
| g | Gravitational acceleration | m/s2 |
| R | Turn radius | m |
What it means
The bank angle required for a coordinated turn depends on speed and turn radius. This formula is derived from the centripetal acceleration requirement. In practice, pilots set bank angle to achieve a desired rate of turn. The bank angle also determines the load factor. Understanding this relation is essential for flight planning, instrument procedures, and for designing turn manoeuvres.
Worked example
Bank Angle for Turn – Two Examples
Real‑World| Parameter | Value |
|---|---|
| V | 100 m/s |
| R | 1000 m |
| Parameter | Value |
|---|---|
| V | 60 |
| R | 600 |
Common mistakes
- Bank angle for coordinated turn: φ = arctan(V² / (g·R)).
- V: True airspeed (m/s).
- R: Turn radius (m).
- g: 9.81 m/s².
- Result in radians – convert to degrees as needed.
Applications
Bank angle for a coordinated turn, φ = arctan(V²/(g·R)), relates the required bank angle to speed and turn radius. It is used for flight planning, air traffic control, and aircraft performance. Engineers use this to determine appropriate bank angles for holding patterns and approach procedures, and to ensure that the aircraft can achieve the required turn performance. By calculating bank angle, aerospace engineers can design flight control laws and assess the feasibility of specific manoeuvres.
- Turn performance analysis for air traffic control
- Hold pattern design and flight planning
- Approach procedures and missed approach turns
- Control system design for bank angle commands
- Human factors and pilot training for coordinated turns
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