Belt & Pulley Speed Ratio Calculator
Solve the belt-drive speed relation for driver speed, driver diameter, driven speed, or driven diameter — with a full step-by-step solution, sensitivity analysis, and a companion belt-speed check.
Driver speed (N₁) ↑
For fixed pulley diameters, driven speed N₂ increases in direct proportion to N₁ — a linear relationship.
Driver diameter (D₁) ↑
A larger driver pulley speeds up the driven shaft proportionally, for a fixed N₁ and D₂ — this is how "step-up" drives are built.
Driven diameter (D₂) ↑
A larger driven pulley slows the output down — N₂ falls inversely with D₂, the classic "bigger pulley spins slower" rule.
Speed ratio
The ratio i = N₁/N₂ = D₂/D₁ depends only on the diameter ratio — the same two pulleys give the same ratio whether the drive runs fast or slow.
How to Use This Calculator
The belt-drive speed relation links four quantities: driver speed N₁, driver diameter D₁, driven speed N₂, and driven diameter D₂. Pick which one you want the tool to solve, then fill in the other three.
1. Choose what to solve for
Pick N₁, D₁, N₂, or D₂ from the "Solve for" buttons — the tool rearranges the formula automatically.
2. Set the driver (motor) speed
Use a standard motor-speed preset (1450 / 1750 / 2900 / 3450 RPM) or type a custom value for N₁.
3. Enter the other known values
Type into the number boxes or drag the sliders for whichever fields are still editable — the greyed-out field is the one being solved.
4. Read the result and steps
The result card shows the solved value plus engineering (×10ⁿ) notation. The "step-by-step solution" panel shows exactly how the number was reached.
5. Switch to Advanced for more depth
Advanced mode adds sensitivity to a ±1% change in each known input, plus a companion belt-speed and ratio check.
6. Copy or reset
Use "Copy result" to grab the solved value, or "Reset" to return to the default worked example.
What This Tool Offers
Built around the classic belt-and-pulley speed relation N₁·D₁ = N₂·D₂, used to size pulleys, sheaves, and sprockets for a target speed change on any single-stage belt or chain drive.
Solve any variable
Go from pulley sizes to output speed, or rearrange the same relation to size a pulley for a target speed — pick the unknown and the tool handles the algebra.
Standard motor-speed presets
Quick presets for common induction-motor sync speeds (1450 / 1750 / 2900 / 3450 RPM), or enter any custom driver speed.
Transparent step-by-step math
Every calculation shows the intermediate cross-multiplication so results can be checked by hand.
Companion drive metrics
Advanced mode also reports the speed ratio and the belt's linear (pitch-line) speed at both pulleys, as a built-in no-slip consistency check.
Built-in sensitivity check
See how much the result moves for a ±1% nudge in each known input — useful for pulley-tolerance and design-margin checks.
Engineering notation & units
Results are shown in both standard and engineering (×10ⁿ) notation with the correct unit attached, so nothing gets lost in translation.
Reference Tables & Unit Conversions
Live converters and lookup tables for the units this calculator uses most — handy when a datasheet gives you diameters in inches or speeds in rad/s or Hz instead of mm and RPM.
Diameter converter
Speed converter
Unit Conversion Factors
| Quantity | From → To | Multiply by |
|---|---|---|
| Diameter | mm → in | ÷ 25.4 |
| Diameter | in → mm | × 25.4 |
| Speed | RPM → rad/s | × π/30 (≈ 0.10472) |
| Speed | rad/s → RPM | × 30/π (≈ 9.5493) |
| Speed | RPM → Hz (rev/s) | ÷ 60 |
| Speed | Hz (rev/s) → RPM | × 60 |
| Belt speed | m/s → ft/min | × 196.85 |
| Belt speed | ft/min → m/s | × 0.00508 |
Standard AC Motor Speeds
Typical synchronous and full-load (with slip) speeds by pole count — handy defaults for the driver speed N₁.
| Poles | Sync @ 50 Hz | Full-load @ 50 Hz | Sync @ 60 Hz | Full-load @ 60 Hz |
|---|---|---|---|---|
| 2 | 3000 RPM | ~2900 RPM | 3600 RPM | ~3450 RPM |
| 4 | 1500 RPM | ~1450 RPM | 1800 RPM | ~1750 RPM |
| 6 | 1000 RPM | ~960 RPM | 1200 RPM | ~1150 RPM |
| 8 | 750 RPM | ~720 RPM | 900 RPM | ~850 RPM |
Common Standard Pulley Diameters
A sample of common stock pulley / sheave pitch diameters, in both metric and imperial.
| mm | in | mm | in |
|---|---|---|---|
| 63 | 2.48 | 200 | 7.87 |
| 71 | 2.80 | 224 | 8.82 |
| 80 | 3.15 | 250 | 9.84 |
| 100 | 3.94 | 315 | 12.40 |
| 125 | 4.92 | 400 | 15.75 |
| 160 | 6.30 | 500 | 19.69 |
Quick Speed-Ratio Reference
For a driver speed of 1450 RPM, the driven speed at common diameter ratios (D₂ : D₁):
| Ratio D₂ : D₁ | Driven speed N₂ (from 1450 RPM) | Effect |
|---|---|---|
| 1 : 1 | 1450 RPM | No change |
| 1.5 : 1 | 967 RPM | Reduction |
| 2 : 1 | 725 RPM | Reduction |
| 2.5 : 1 | 580 RPM | Reduction |
| 3 : 1 | 483 RPM | Reduction |
| 1 : 2 | 2900 RPM | Step-up |
| 1 : 3 | 4350 RPM | Step-up |
What Is the Belt & Pulley Speed Ratio?
In a simple belt (or chain) drive, a driver pulley of diameter D₁ turning at speed N₁ transmits motion to a driven pulley of diameter D₂ turning at speed N₂. Because the belt runs at the same linear (pitch-line) speed on both pulleys — assuming no slip — the product of speed and diameter must be the same on each side of the drive. That single fact is the whole basis of pulley sizing: pick any two of speed-in, speed-out, or the two diameters, and the fourth is fixed.
Belt & Pulley Speed Ratio Formula
The defining relationship is:
Because the equation is purely algebraic, it can be rearranged directly to solve for any of the four quantities:
D₁ = N₂·D₂ / N₁
N₂ = N₁·D₁ / D₂
D₂ = N₁·D₁ / N₂
Belt & Pulley Speed Ratio Variables
| Symbol | Meaning | Typical range |
|---|---|---|
| N₁ | Driver pulley rotational speed | ~500 – 3,600 RPM (typical motor speeds) |
| D₁ | Driver pulley diameter (pitch line) | ~20 – 500 mm |
| N₂ | Driven pulley rotational speed | ~50 – 3,600 RPM, depending on ratio |
| D₂ | Driven pulley diameter (pitch line) | ~20 – 1,000 mm |
How Each Variable Affects the Ratio
Speed and diameter trade off directly: for fixed pulley sizes, driven speed rises and falls in step with driver speed. For a fixed driver setup, making the driven pulley bigger always slows the output down, and making it smaller always speeds it up — the speed ratio i = N₁/N₂ = D₂/D₁ depends only on the ratio of the two diameters, never on their absolute size.
Belt & Pulley Calculation Examples
N₂ = N₁·D₁ / D₂ = 1450 × 100 / 250
N₂ = 580 RPM
D₂ = N₁·D₁ / N₂ = 1750 × 80 / 700
D₂ = 200 mm
Frequently Asked Questions
It assumes a single continuous belt (or chain) running with no slip, so both pulleys share exactly the same pitch-line linear speed. Real belts slip slightly under load, which is why the calculated ratio is a close approximation rather than an exact guarantee.
It's already an inverse proportion written as a cross-product: speed and diameter move oppositely so the belt's linear speed stays constant, which is exactly what N·D held equal on both sides expresses.
The pitch diameter — the effective diameter at the belt's centerline — gives the most accurate result. Outer diameter is a common, slightly less precise stand-in, especially for V-belts and toothed belts where the pitch line sits below the outer edge.
The same cross-multiplied form applies to chain sprockets and meshing gears if you swap diameter for tooth count (N₁·T₁ = N₂·T₂), since tooth count plays the same role as effective diameter in those positive-drive systems.
Solve each belt stage separately with this calculator, then multiply the individual stage ratios together to get the overall speed ratio across the whole compound drive.
Yes — the ratio is dimensionless as long as both diameters use the same unit. Enter both D₁ and D₂ in inches (or any consistent unit) and the solved speed will still be correct.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| N1, N2 | Rotational speeds of pulley 1 and pulley 2 | RPM |
| D1, D2 | Diameters of pulley 1 and pulley 2 | mm |
What it means
In a belt drive, the linear speed of the belt must be identical on both the driver and driven pulleys. This leads to the relationship N₁ D₁ = N₂ D₂, where N is rotational speed (e.g., rpm) and D is the pulley diameter. This formula assumes no slip between the belt and pulley. It shows that the speed ratio is inversely proportional to the diameter ratio. If the driven pulley is larger, it turns slower but produces higher torque. This principle is used in various machines, such as conveyor systems, automotive accessories (alternator, water pump), and industrial drives. The belt tension and friction determine the torque transmission capacity. In practice, slip may occur, which affects the speed ratio. The formula is also applicable to chain drives and gear trains (with teeth count instead of diameters). Proper selection of pulley sizes is crucial for achieving desired output speeds and torque. This relationship is fundamental in mechanical power transmission design.
Worked example
Belt and Pulley – Two Examples
Real‑World| Parameter | Value |
|---|---|
| N1 | 1500 RPM |
| D1 | 100 mm |
| D2 | 200 mm |
| Parameter | Value |
|---|---|
| N1 | 500 RPM |
| D1 | 200 mm |
| D2 | 100 mm |
Common mistakes
- Speed N in rpm: Not rad/s – use rpm consistently.
- Diameter D: Use the pitch diameter of the pulley (or outer diameter if belt thickness ignored).
- Belt slip: The formula assumes no slip; in reality, slip reduces the speed ratio.
- Direction of rotation: The product equality holds regardless of direction, but ensure you assign N₁ and N₂ correctly.
- Units: Both diameters in same units (mm, m, etc.) – the ratio cancels.
Applications
In a belt and pulley system, the product of speed and diameter is constant, ensuring the belt travels at the same linear speed. This principle is used to transmit power between shafts at different speeds and distances. It is common in machinery, such as in conveyors, fans, compressors, and textile equipment. The speed ratio allows engineers to select appropriate pulley diameters to achieve desired output speeds. Belt drives are favoured for their simplicity, low cost, and overload protection. By applying this relationship, engineers can design efficient power transmission systems that accommodate space constraints and speed requirements.
- Power transmission between shafts in machinery
- Conveyor belt and material handling systems
- Automotive accessory drives (alternator, water pump)
- HVAC fan and blower drives
- Textile and printing machinery
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