Turn rate and turn radius

Turn Rate and Turn Radius

Turn rate (ω) and turn radius (r) are the two key kinematic quantities of a turn.

Formulas

Turn radius

r = V² / (g · tan φ)

with:

• V = TAS (m/s)
• g = 9.81 m/s²
• φ = bank angle

Turn rate

ω = g · tan φ / V (rad/s)

or in °/s: ω = (g · tan φ / V) · 180/π

Standard rate turn (rate 1 turn)

Standard rate turn = 3°/second = 360°/2 minutes.

Bank angle for standard rate turn (rule of thumb):

φ ≈ V/10 + 7 (V in kt, φ in degrees)

More precisely: tan(φ) ≈ V/200 (V in kt, approximately).

Worked examples

Example 1: Cessna 172 at 30° bank

• V = 100 kt = 51.4 m/s
• φ = 30°, tan(30°) = 0.577
• r = (51.4)² / (9.81 × 0.577) = 2642 / 5.66 = 467 m ≈ 0.25 NM
• ω = (9.81 × 0.577) / 51.4 = 0.110 rad/s = 6.3°/s
• 360° in 360/6.3 = 57 seconds

Example 2: Cessna 172 at 45° bank

• V = 100 kt = 51.4 m/s
• φ = 45°, tan(45°) = 1.0
• r = (51.4)² / (9.81 × 1.0) = 269 m ≈ 0.15 NM (~half as tight as 30°)
• ω = (9.81 × 1.0) / 51.4 = 0.191 rad/s = 10.9°/s
• 360° in 33 seconds

Example 3: Cessna 172 at 60° bank

• V = 100 kt = 51.4 m/s, φ = 60°, tan(60°) = 1.73
• r = (51.4)² / (9.81 × 1.73) = 156 m ≈ 0.08 NM
• ω = (9.81 × 1.73) / 51.4 = 0.330 rad/s = 18.9°/s
• 360° in 19 seconds

Trade-off: bank vs stall speed

→ Steeper bank = tighter turn, but higher stall risk.

Operational significance

Pattern turns

VFR pattern: usually 20°-30° bank.

• At 30° and 90 kt: turn radius about 400 m → fits typical pattern.
• At 60° the pattern would be far too tight — dangerously close to the runway.

Cross-country

Cross-country: 15°-20° bank for comfort and visibility.

• At 15°: r = 870 m, ω = 3.2°/s — gentle.

Holding pattern

Holding requires standard rate turn (rate 1) = 3°/s.

• At 100 kt: 17° bank → 1 turn (180°) in 1 minute.

Steep turns (exercise / aerobatic)

45° bank = standard for steep-turn exercise in skill test.

• Pilot demonstrates altitude hold ±100 ft, airspeed ±10 kt.

Effect of speed

Doubling V at same bank:

• Radius quadrupled (r ∝ V²).
• Turn rate halved (ω ∝ 1/V).

→ Slow = tight turn, fast = wide turn.

Speed-bank trade-off relationships

"Same g, different V"

At same load factor n (same bank):

• Double speed → quadruple radius.

"Same radius, different V"

At same radius:

• Double speed → tan φ quadrupled → φ steeper.

"Same turn rate, different V"

At same turn rate ω (e.g. rate 1):

• Double speed → tan φ doubled → φ steeper.

Practical limit

Max bank at normal PPL: 45° (standard skill test).

Above 60° you should:

• Have experience.
• Altitude margin above ground (at least 1500 ft AGL).
• Speed reserve above Vs(n) (at least 20 kt above Vs(n)).
• Aerobatic certification of the aircraft (for >60° continuous).