Internal Gear Calculator: Ratios & RPM

This internal gear calculator works out the gear ratio and output speed (RPM) for a pair of meshing gears where one gear has teeth on the inside of a ring. You enter the number of teeth on the sun gear (the smaller central gear) and the ring gear (the large internal gear), and the calculator returns the gear ratio and the output RPM for a given input speed. It is most often used for planetary gear sets, ring-and-pinion setups, and internal gear trains in machinery.

Formula

The basic internal gear ratio (for a simple internal gear pair, with the ring fixed and the sun gear driving) is:

Gear Ratio = N_ring / N_sun

Output speed (when the ring is fixed and the sun gear is the input) is:

RPM_out = RPM_in × (N_sun / N_ring)

Where:

• N_ring = number of teeth on the internal (ring) gear (unitless)
• Nsun = number of teeth on the sun gear (unitless)
• RPM_in = input rotational speed (revolutions per minute)
• RPM_out = output rotational speed (revolutions per minute)

This formula assumes the ring gear is held stationary and the sun gear is the input. If the ring gear rotates instead, the ratio changes—see the section on planetary gear sets below.

Worked Example

Using the default values from the calculator: sun gear has 20 teeth, ring gear has 80 teeth, and input speed is 1000 RPM.

1. Calculate the gear ratio: 80 / 20 = 4.00 : 1
2. Calculate the output speed: 1000 × (20 / 80) = 1000 × 0.25 = 250 RPM

The calculator shows a ratio of 4.00:1 and an output of 250.0 RPM, matching the manual calculation. This means the sun gear turns four times for every one turn of the ring gear (when the ring is fixed).

When to Use an Internal Gear Calculator

Internal gears are used in many mechanical systems where a compact, high-ratio speed reduction is needed. Common applications include:

• Planetary gearboxes (also called epicyclic gears) in automatic transmissions, electric screwdrivers, and wind turbines.
• Differential gears in vehicles, where an internal ring gear (the crown wheel) meshes with pinion gears.
• Rotary tables and indexing heads in machine tools.
• Winches and hoists that need a large torque multiplication in a small space.

Use this calculator when you know the tooth counts and need to find the ratio or output speed. If you are designing a gear train, you might also need to consider the number of planet gears and their tooth counts—this calculator handles only the simple two-gear case.

How to Choose the Right Tooth Counts

Choosing tooth counts is a balance between ratio, strength, and manufacturability. Here are the main factors:

• Ratio needed: A higher ring-to-sun ratio gives a higher reduction (more torque, less speed). For a 10:1 reduction, you might use a 20-tooth sun and a 200-tooth ring, but that ring may be large.
• Minimum teeth: For standard 20° pressure angle gears, the minimum number of teeth to avoid undercutting is about 17 for external gears, but internal gears can have fewer teeth on the sun if the mesh is properly designed. A common rule is to keep the sun gear above 12 teeth for strength.
• Tooth interference: The difference in teeth between ring and sun must be enough to avoid interference. A rule of thumb: the difference should be at least 8 teeth for standard gears.
• Space constraints: A larger ring gear takes more physical space. If space is tight, you might use a multi-stage planetary gearbox instead of a single large ring.

In practice, gear designers also check the contact ratio and bending stress, but for a quick estimate, the tooth count ratio is the starting point.

Common Mistakes and Misconceptions

A frequent error is mixing up the driving and driven gear. In an internal gear pair, the smaller gear (sun) usually drives the larger ring gear, but if the ring gear is the input, the ratio is inverted. Always confirm which gear is fixed and which is the input.

Another misconception: the gear ratio is not simply the ratio of diameters. For gears, the ratio is based on the number of teeth, which is proportional to pitch diameter, but the pitch diameter depends on the module (or diametral pitch). Two gears with the same tooth count but different module will have different diameters but the same ratio.

Also, do not forget that the direction of rotation reverses with an external gear pair, but with an internal gear pair, the sun and ring rotate in the same direction. This is a key difference from external gears.

Limits of This Calculator

This calculator only handles a simple two-gear internal mesh with the ring fixed. It does not account for:

• Planet gears (in a planetary set, the ratio depends on the planet carrier, which is not included here).
• Tooth profile, pressure angle, or module—these affect strength and noise, not the basic ratio.
• Friction, efficiency, or backlash.
• Torque or power—you need to know the input torque and efficiency to get output torque.

For a full planetary gear set, the ratio is different. For example, with a fixed ring gear, sun input, and planet carrier output, the ratio is (N_ring + N_sun) / N_sun. That is a more complex formula that this simple calculator does not cover.

Frequently Asked Questions

What is the difference between internal and external gears?

External gears have teeth on the outside of a cylinder and mesh with other external gears, rotating in opposite directions. Internal gears have teeth cut on the inside of a ring, and the meshing gear rotates in the same direction. Internal gears also allow a more compact design because the center distance is smaller for a given ratio.

How do I calculate the center distance for internal gears?

The center distance for internal gears is (N_ring – N_sun) × module / 2. You need the module (or diametral pitch) to get a physical distance. This calculator does not include module because it is a separate input.

Can I use this calculator for a planetary gear set?

No, this calculator is only for a simple internal gear pair. For a planetary set, you need to specify which component is fixed (sun, ring, or carrier) and include the planet gear teeth. The ratio formulas are different for each configuration.

What is a good number of teeth for a sun gear?

For standard 20° pressure angle gears, a sun gear with fewer than 17 teeth may have undercutting, which weakens the tooth. Many designs use 20 to 40 teeth for the sun. However, with profile shift or special pressure angles, you can go lower. Always check the manufacturer’s data for the specific gear.

Does the direction of rotation change with internal gears?

No, with an internal gear pair, the sun gear and ring gear rotate in the same direction. This is opposite to external gears, where the direction reverses. This is an important consideration when designing a gear train.

What is the efficiency of internal gears?

Internal gear pairs are generally more efficient than external gears because there is less sliding friction. Efficiencies of 98% to 99% per mesh are typical for well-lubricated gears, but this depends on the tooth profile, speed, and lubrication. This calculator does not include efficiency.

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in teeth_sun | Sun gear teeth | - | 20 | 1
in teeth_ring | Ring gear teeth | - | 80 | 2
in rpm_in | Input speed | RPM | 1000 | 0
out ratio | Gear ratio (sun:ring) | :1 | teeth_ring / teeth_sun | 2
out rpm_out | Output speed | RPM | rpm_in * (teeth_sun / teeth_ring) | 1