Rotor Solidity Ratio Calculator
Solve the rotor solidity relation σ = (Nb · c) / (π · R) for any of its four unknowns, with step-by-step math, a live rotor disc schematic, and full companion rotorcraft metrics including thrust coefficient, blade loading, and disc loading.
Blade count and chord scale linearly
Doubling the number of blades doubles solidity. Doubling the chord also doubles solidity. These are the two levers designers can pull to change blade area without changing rotor size — but adding blades costs weight and complexity, while adding chord costs profile power.
Radius has the strongest effect
σ ∝ 1/R. A rotor with the same blade area but twice the radius has half the solidity. This is why large, slow-turning rotors (heavy lift, human-powered) have very low solidity — the disc is huge and the blades are slender.
Solidity sets thrust coefficient
Blade element theory gives CT/σ ≈ (CL/2) at the 3/4 radius. So for a given lift coefficient, solidity sets the thrust coefficient. Low-solidity rotors need high lift coefficients (near stall) to produce thrust; high-solidity rotors produce the same thrust at low CL.
Profile power ∝ σ
Profile power is roughly P0 = σ Cd0 ρ A (ΩR)³ / 8. It scales linearly with solidity. This is why very high solidity is inefficient — you pay a profile-drag penalty on every blade element. Modern designs sit at 0.08–0.12 as a compromise.
How to Use This Calculator
The rotor solidity relation links four quantities: the solidity, the number of blades, the chord, and the rotor radius. Pick which one you want the tool to solve, then fill in the others.
1. Choose what to solve for
Pick σ, Nb, c, or R — the tool rearranges the relation automatically for whichever you choose.
2. Use realistic presets
Solidity presets span 0.05–0.15, blade counts from 2 to 8, chord from 0.15 to 0.65 m, and radius from 3.85 m (R22) to 9.15 m (CH-47).
3. Read the result and steps
The result card shows the solved quantity with engineering notation. The step-by-step panel walks through the arithmetic exactly as the formula evaluates it.
4. Check the rotor disc schematic
The top-view diagram shows the rotor disc, the correct number of blades, the radius arrow, and the chord dimension on one blade. All update live with your inputs.
5. Switch to Advanced for depth
Advanced mode adds a chart of σ vs. radius, a 1% sensitivity check on every input, and companion metrics including blade area, disc area, aspect ratio, and typical thrust coefficient at a reference CL.
6. Cross-check with the reference table
The reference section lists solidity for typical helicopters and gives a lookup grid across the chord–radius envelope.
What This Tool Offers
Built around the classical rotor solidity definition used throughout rotorcraft design and blade element analysis.
Four-way solving
Solve for solidity, number of blades, chord, or radius — whichever the problem gives you.
Live rotor disc schematic
A top-view diagram showing the exact blade count, radius arrow, and chord dimension — all updating live.
Dual comparison charts
σ vs. chord at three blade counts, plus σ vs. radius at three chord values.
Companion rotorcraft metrics
Advanced mode reports blade area, disc area, blade aspect ratio, thrust coefficient at reference CL, and typical profile power fraction.
Built-in sensitivity check
See how much σ moves for a 1% change in each input. The power-law exponents (+1, +1, −1) are shown explicitly.
Reference tables
A lookup table of solidity across typical chord and radius combinations, plus typical values for common helicopter classes.
Reference Tables
Solidity across typical rotor envelopes, plus example configurations for common helicopter classes.
Solidity quick lookup (Nb = 4)
σ = (Nb · c) / (π · R) for various chords and rotor radii.
| Chord c (m) | R = 3 m | R = 5 m | R = 8 m | R = 12 m | R = 16 m |
|---|
Typical Values by Helicopter Class
Representative values at maximum takeoff weight. Actual figures vary with configuration and operating condition.
| Class | Example | Nb | c (m) | R (m) | σ |
|---|---|---|---|---|---|
| Light single | Robinson R22 | 2 | 0.18 | 3.85 | 0.0298 |
| Light single | Bell 206 JetRanger | 2 | 0.32 | 5.08 | 0.0401 |
| Intermediate single | Airbus H125 | 3 | 0.35 | 5.53 | 0.0604 |
| Twin medium | Sikorsky UH-60 | 4 | 0.53 | 8.18 | 0.0825 |
| Attack | AH-64 Apache | 4 | 0.53 | 7.32 | 0.0921 |
| Twin heavy | Boeing CH-47 | 3 | 0.81 | 9.15 | 0.0845 |
Variable Reference
| Symbol | Meaning | Typical range | Units |
|---|---|---|---|
| σ | Rotor solidity ratio | 0.05 – 0.15 | — |
| Nb | Number of blades | 2 – 8 | — |
| c | Blade chord | 0.15 – 0.65 | m |
| R | Rotor radius | 3 – 16 | m |
| A | Disc area = πR² | 30 – 800 | m² |
| Ab | Total blade area = Nb·c·R | 2 – 60 | m² |
What Does the Rotor Solidity Ratio Measure?
Rotor solidity is the fraction of the rotor disc that is covered by blade planform area. It is defined as the ratio of total blade area (Nb · c · R) to disc area (π R²), which simplifies to σ = (Nb · c) / (π · R). A solidity of 0.10 means that 10% of the disc is blade and 90% is open air. Solidity is the single most important geometric parameter in blade element theory: it relates thrust coefficient to blade lift coefficient, sets the profile-power fraction, and constrains the blade loading at which the rotor can operate.
Rotor Solidity Formula
The defining relationship is:
It can also be written as the ratio of total blade planform area to disc area:
Rearranged to solve for the number of blades, chord, or radius:
Rotor Solidity Variables
| Symbol | Meaning | Notes |
|---|---|---|
| σ | Rotor solidity ratio | Dimensionless; typically 0.05–0.15 for helicopters |
| Nb | Number of blades | Integer ≥ 2 for practical rotors (1 would be unbalanced) |
| c | Blade chord | Assumed constant along the span in this simplified relation |
| R | Rotor radius | Distance from hub to blade tip |
How Each Parameter Moves the Solidity
The relation is linear in Nb and c, and inverse-linear in R. Adding a blade increases solidity by exactly the same proportion as increasing chord by 50% would (if going from 2 to 3 blades). Increasing radius always reduces solidity, which is why large heavy-lift rotors can use wide chords and still achieve moderate solidity. Typical design values sit between 0.07 and 0.12 for helicopters; values below 0.05 usually indicate a very lightly loaded rotor (human-powered, solar), and values above 0.15 are rare outside of specific compound or high-thrust applications.
Rotor Solidity Calculation Examples
Nb · c = 4 × 0.35 = 1.400 m
π · R = π × 4.95 = 15.551 m
σ = 1.400 / 15.551 = 0.0900
Blade area = Nb · c · R = 1.4 × 4.95 = 6.93 m²
Disc area = π R² = π × 24.50 = 76.97 m²
c = σ · π · R / Nb = 0.08 × π × 5.5 / 3
= 0.08 × 17.279 / 3 = 1.382 / 3 = 0.4607 m
Frequently Asked Questions
For conventional helicopters, solidity ranges from about 0.05 to 0.15. Light singles like the Robinson R22 sit around 0.03–0.05. Medium twins like the UH-60 are 0.08–0.10. Heavy lift and compound helicopters reach 0.10–0.15. Human-powered helicopters, which need very large, lightly loaded rotors, use solidity below 0.02. The value depends on the trade-off between thrust capability (higher σ helps) and profile power (lower σ helps).
Blade element theory relates the thrust coefficient CT to the blade section lift coefficient CL through solidity: CT ≈ (σ · CL) / 2 at the 3/4 radius. This means that for a given thrust, the required blade lift coefficient is inversely proportional to solidity. A low-solidity rotor must operate at high CL (near stall), while a high-solidity rotor reaches the same thrust at low CL. Solidity therefore directly controls the stall margin and the maximum thrust the rotor can produce.
Profile power — the power required to overcome the drag of the blades themselves, independent of the thrust they produce — scales linearly with solidity. The formula is P0 = (σ · Cd0 · ρ · A · (ΩR)³) / 8. Doubling solidity doubles profile power. This is why very high solidity is inefficient: you pay a drag penalty on every blade element, even when the rotor is producing zero net thrust. Modern helicopter rotors balance this against the thrust benefit and typically sit at 0.08–0.12.
The relation σ = (Nb · c) / (π · R) assumes a rectangular blade with constant chord. In practice, many modern blades are tapered — wider at the root and narrower at the tip — to reduce tip Mach effects and improve aerodynamic efficiency. For a tapered blade, you can compute a local solidity σ(r) = Nb · c(r) / (π · R), which varies along the span. The global solidity is then the area-weighted average. The simplified form here gives the mean solidity, which is what appears in most textbook blade element formulations.
Blade count is a design trade-off. Fewer blades mean lower weight, lower cost, and simpler hub design, but higher vibration and higher blade loading (higher CL for the same thrust). More blades distribute the load, reduce vibration and noise, and allow lower blade loading, but increase weight, drag, and hub complexity. Two-blade rotors (like the R22 and Bell 206) can achieve reasonable solidity with a wide chord. Four-to-six-blade rotors are common on modern medium helicopters. Very high blade counts (7–8) appear on high-speed compounds and next-generation designs.
Disc loading (DL = W/A) and solidity are two independent parameters that together define the rotor's operating point. Disc loading tells you how hard the rotor is working; solidity tells you how much blade area it has to work with. A rotor with high disc loading and low solidity must operate at very high blade lift coefficients (near stall), which is inefficient and dangerous. A rotor with low disc loading and high solidity operates at very low CL, which is safe but wastes profile power. Designers tune both to land in an efficient, safe envelope.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| σ | Rotor solidity | |
| N_b | Number of blades | |
| c | Blade chord | m |
| R | Rotor radius | m |
What it means
Solidity is the ratio of the total blade area to the disk area. It influences the thrust coefficient and the blade loading. Higher solidity allows more thrust for a given disk loading but increases profile drag. Solidity is a key design parameter for rotors, affecting performance and noise. It is used in the design of both helicopters and propellers. Understanding solidity is essential for rotor aerodynamics and for sizing blades.
Worked example
Rotor Solidity – Two Examples
Real‑World| Parameter | Value |
|---|---|
| N_b | 4 |
| c | 0.3 m |
| R | 6 m |
| Parameter | Value |
|---|---|
| N_b | 5 |
| c | 0.28 |
| R | 5 |
Common mistakes
- Rotor solidity ratio: σ = (N_b · c) / (π·R).
- N_b: Number of blades.
- c: Mean blade chord (m).
- R: Rotor radius (m).
- Dimensionless – indicates blade area fraction.
Applications
Rotor solidity, σ = (N_b·c)/(π·R), is the ratio of total blade area to rotor disk area. It affects thrust, power, and vibration characteristics. High solidity rotors can produce more thrust but have higher drag and weight. Engineers use solidity to design rotors for specific missions, balancing thrust capability with efficiency. By selecting appropriate solidity, aerospace engineers can optimise rotor performance for hover, forward flight, and manoeuvring, while also considering dynamic loads and noise.
- Helicopter rotor design and optimisation
- Thrust and power capacity trade‑offs
- Rotor structural design and blade count selection
- Vibration and dynamic stability considerations
- Design of rotors for different vehicle types (helicopters, tiltrotors)
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