Turn Radius
Radius of a coordinated banked turn.
Formula
Variables
Example
Steeper bank, tighter turn
How it works
This formula gives the horizontal turning radius of an aircraft in a coordinated, level banked turn. It comes from balancing the horizontal component of lift, which supplies the centripetal force for turning, against the required circular-motion force; the steeper the bank, the larger the sideways lift component and the smaller the radius. Because the radius scales with v^2, even a modest increase in speed makes the turn much wider.
When to use it
Use this when analyzing coordinated turns in flight mechanics, especially to estimate how much space an aircraft needs to make a given turn at a certain speed and bank angle. It is also useful in performance calculations, instrument flight planning, and understanding why low-speed, steep turns are easier to fit into tight airspace than fast, shallow ones.
Worked example, step by step
An aircraft is flying at 80 m/s in a coordinated banked turn with bank angle \(\phi = 30^\circ\). Step 1: Compute \(\tan\phi\): \(\tan 30^\circ \approx 0.577\) Step 2: Square the speed: \(v^2 = 80^2 = 6400\) Step 3: Compute the denominator: \(g\tan\phi = 9.81 \times 0.577 \approx 5.66\) Step 4: Apply the formula: \(r = \frac{6400}{5.66} \approx 1131\text{ m}\) Final answer: The turn radius is about \(1.13 \times 10^3\) m, or roughly 1.13 km.
Common mistakes to avoid
- Using the bank angle in degrees directly without taking the tangent
- Forgetting that speed must be squared, which makes the radius grow very quickly with speed
- Applying the formula to an uncoordinated or climbing/descending turn without checking the assumptions
Did You Know?
A faster aircraft needs a wider turn, which is why jets sweep through the sky in gentle arcs.
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