What is a drag polar?

A drag polar is the relationship between an aircraft's lift coefficient CL and drag coefficient CD, usually plotted as CL against CD. The line from the origin tangent to the curve touches it at the maximum lift-to-drag point.

CD0 is the zero-lift (parasite) drag coefficient: the drag coefficient the parabolic polar predicts when CL = 0. It covers skin friction, form and interference drag, while k·CL² adds the lift-dependent induced drag.

What is the drag polar equation?

The parabolic drag polar is CD = CD0 + k·CL², where CD0 is the zero-lift (parasite) drag coefficient and k·CL² is the induced drag term. This equation accurately models subsonic aircraft drag over most of the CL operating range.

How is k related to wing geometry?

The induced drag factor k = 1 / (π·e·AR), where e is the Oswald span efficiency factor (typically 0.7–0.85 for a whole aircraft) and AR is the wing aspect ratio. A higher AR or higher e gives a smaller k and therefore less induced drag. Example: AR = 8 and e = 0.80 give k = 1/(π × 0.80 × 8) = 0.0497.

How do I estimate the Oswald efficiency factor e?

Raymer's empirical fits are widely used in conceptual design: for straight wings e = 1.78·(1 − 0.045·AR^0.68) − 0.64; for leading-edge sweep above 30° e = 4.61·(1 − 0.045·AR^0.68)·(cos Λ_LE)^0.15 − 3.1. AR = 8 with a straight wing gives e ≈ 0.81. Flight-test or wind-tunnel data, fitted with the curve-fit box on this page, are better when available.

What are typical values for CD0?

For a clean subsonic jet transport, CD0 is roughly 0.015–0.025. General aviation piston aircraft typically range from 0.025–0.040. Gliders can be as low as 0.010–0.015, while a pod-and-boom UAV might be 0.030–0.060.

What is the minimum drag point?

For the parabolic model, maximum positive L/D occurs where the two modeled drag terms are equal: CD0 = k·CL². This gives CL = sqrt(CD0/k), CD = 2·CD0 and (L/D)max = 1/(2·sqrt(CD0·k)). Calling this the minimum-CD point is incorrect: CD itself is smallest at CL = 0.

What is the maximum lift-to-drag ratio?

Within this model, maximum positive L/D is (L/D)max = 1/(2·sqrt(CD0·k)). Translating that coefficient condition into aircraft speed or a range/endurance optimum requires additional aircraft, atmosphere and propulsion assumptions not included here.

What is the minimum drag formula using CD0 and k?

In steady level flight lift equals weight W, so drag D = W ÷ (L/D). Minimum drag, and minimum thrust required, therefore occurs at maximum L/D: Dmin = 2·W·sqrt(CD0·k), at CL = sqrt(CD0/k) where induced and parasite drag are equal. The matching true airspeed is V = sqrt(2W/(ρ·S·CL)).

Which CL gives best endurance or range on a parabolic drag polar?

Maximum CL^1.5/CD (minimum power required, propeller endurance) occurs where k·CL² = 3·CD0, so CL = sqrt(3·CD0/k) and CD = 4·CD0; it is flown at 0.76 times the max-L/D speed. Maximum CL^0.5/CD (jet range) occurs where k·CL² = CD0/3, so CL = sqrt(CD0/(3k)) and CD = 4·CD0/3, at 1.32 times the max-L/D speed. At both points L/D = 0.866 × (L/D)max. Maximum L/D itself gives propeller range, jet endurance and best glide.

How do I find CD0 and k from wind-tunnel or flight data?

Plot CD against CL². For a parabolic polar the points lie on a straight line: the intercept is CD0 and the slope is k. A least-squares fit of several (CL, CD) pairs, as in the curve-fit box on this page, gives both plus R². Then e = 1/(π·AR·k). Leave out points near stall and the drag bucket of laminar airfoils, where the polar is not parabolic.

Does this tool handle compressibility effects?

No. The parabolic drag polar is a subsonic, incompressible model. Near and above Mach 0.6, wave drag increases CD0 significantly and the polar becomes non-parabolic. For transonic analysis, use drag-divergence corrections or more detailed CFD data.

Drag Polar Calculator

Enter CD0 and k (or AR and Oswald e) to get CD and L/D at any CL, the max L/D, minimum-power and jet-range points, plus speeds, drag and power for your weight, wing area and altitude. Paste wind-tunnel or flight data to fit CD0 and k.

A drag polar is the curve linking an aircraft's drag coefficient to its lift coefficient, usually modeled as CD = CD0 + k·CL², where CD0 is zero-lift drag and k = 1/(π·e·AR).

Drag Polar Parameters

CD = CD0 + k · CL²

Zero-Lift Drag Coefficient, CD₀

Parasite drag at zero lift. Jet transport: 0.015–0.025.

Induced Drag Factor, k (= 1/π·e·AR)

Typical range 0.03–0.12. Lower k = less induced drag.

Derive k from aspect ratio and Oswald efficiency

Aspect ratio, AR

Oswald efficiency, e

LE sweep, Λ (°)

AR and e are dimensionless.

Query: Lift Coefficient, CL

Compute CD and L/D at this specific CL value.

Polar Range: CL_max

Upper CL limit for the plotted curve; also used for the stall speed.

Flight Condition (optional)

Weight (kg)

Wing area (m²)

Altitude (m)

ISA standard day, pressure altitude up to 20 km (65,600 ft). Leave weight blank to skip speeds.

CD at CL = 0.50

—

Drag coefficient

Lift-to-Drag Ratio (L/D)

—

Max L/D

—

Best aerodynamic efficiency

At CL_{L/D max} / CD

— / —

Optimum Points & Flight Speeds

Speeds are true airspeed in steady level flight (lift = weight).

Point	CL	CD	L/D	Speed	Drag	Power req.

Drag Polar Curve

CL vs CD

Click "Calculate & Plot" to display the drag polar curve.

Fit CD0 and k From Measured Data

Paste wind-tunnel or flight-test pairs, one “CL, CD” per line. The tool fits CD = CD0 + k·CL² by least squares on CD versus CL² (intercept = CD0, slope = k).

CL, CD pairs

Points used—

CD0 (intercept)—

k (slope)—

R²—

Oswald e at AR above—

(L/D)max from fit—

Drag Polar Optimum Points

Condition	CL	CD	Speed vs V(L/D max)	Used for
Max L/D (min drag)	√(CD0/k)	2·CD0	1.000	Best glide, prop range, jet endurance
Max CL^1.5/CD (min power)	√(3·CD0/k)	4·CD0	0.760	Prop endurance, min sink
Max CL^0.5/CD	√(CD0/(3k))	4/3·CD0	1.316	Jet range

Worked example: CD0 = 0.020, k = 0.050. (L/D)max = 1/(2√(0.020 × 0.050)) = 15.81 at CL = 0.632, CD = 0.040. A 1,100 kg aircraft with 16.2 m² of wing at sea level (ρ = 1.225 kg/m³) flies that point at V = √(2 × 10,787 N / (1.225 × 16.2 × 0.632)) = 41.5 m/s (80.6 kt), with drag W ÷ (L/D) = 682 N and power required 28.3 kW. Minimum power is at CL = 1.095, 31.5 m/s and 24.8 kW.

Sources: J. D. Anderson, Aircraft Performance and Design (parabolic polar optimum conditions); D. P. Raymer, Aircraft Design: A Conceptual Approach (Oswald efficiency estimates); ISA atmosphere per ICAO Doc 7488. Speeds ignore compressibility, ground effect and propulsive efficiency.

Typical Drag Polar Parameters by Aircraft Type

Aircraft Type	CD0	k	Max L/D (approx.)
High-performance glider	0.010	0.018	37.3
Jet transport (clean)	0.020	0.042	17.3
General aviation (piston)	0.030	0.055	12.3
Fighter jet (subsonic, clean)	0.014	0.140	11.3
Fixed-wing UAV	0.040	0.070	9.4

Values are representative approximations for conceptual design. Actual polars depend on wing geometry, Reynolds number, and configuration.

Quick answer

A drag polar is the curve linking an aircraft's drag coefficient to its lift coefficient, usually modeled as CD = CD0 + k·CL², where CD0 is zero-lift drag and k = 1/(π·e·AR). Maximum L/D occurs at CL = √(CD0/k), where CD = 2·CD0.

Example: CD0 = 0.020, k = 0.050, CL = 0.5: CD = 0.020 + 0.050 × 0.25 = 0.0325, so L/D = 15.38; maximum L/D is 15.81 at CL = 0.632 and CD = 0.040.

Enter CD0 and k, or derive k from aspect ratio, to plot your polar and get flight speeds.

Summary

The drag polar calculator computes total drag coefficient CD for a lift coefficient CL using the symmetric parabolic model CD = CD0 + k·CL². CD0 is the zero-lift drag coefficient and k = 1/(π·e·AR) is the induced-drag factor; the Oswald factor e can be estimated with Raymer's empirical formulas. The tool finds the three classic optimum points (maximum L/D, maximum CL^1.5/CD for minimum power and propeller endurance, maximum CL^0.5/CD for jet range), turns them into true airspeed, drag and power required for a given weight, wing area and ISA altitude, and fits CD0 and k to measured (CL, CD) pairs by least squares.

How it works

Enter the zero-lift drag coefficient CD0 (parasite drag at CL = 0).
Enter the induced drag factor k (often written as 1 / (π·e·AR), where e is Oswald efficiency and AR is aspect ratio).
Optionally enter the maximum CL for the polar range (default 1.8); it is also used for the stall speed.
Enter a specific CL value to compute the corresponding CD point.
Click Calculate to get CD, L/D, the max L/D, minimum-power and jet-range points, and the polar curve.
Optionally enter weight, wing area and altitude (SI or US units) to convert each point into speed, drag and power required using the ISA atmosphere.
Paste measured (CL, CD) pairs to fit CD0 and k by linear least squares on CD versus CL².

Use cases

Computing drag coefficient at any lift condition during aircraft conceptual design.
Plotting the full drag polar curve for performance analysis.
Finding the positive-CL maximum lift-to-drag ratio within the parabolic model.
Verifying drag polar models against wind-tunnel or flight-test data.
Teaching aerodynamics and aircraft performance fundamentals.
Finding best-glide, minimum-power and best-range speeds for a given weight and altitude.
Fitting CD0 and k to wind-tunnel or flight-test data.
Comparing polars for different Oswald efficiency factors or aspect ratios.
Supporting UAV and glider performance trade studies.

Frequently Asked Questions

Last updated: 2026-10-09 · Reviewed by Nham Vu