Torque from Force and Radius Calculator + Formula, Examples & Applications

When a hinged lid, gate, or arm won’t move, or the actuator stalls at one end, it’s usually not the actuator’s fault. The mounting geometry is almost always the problem. Torque depends on where you attach the actuator, the distance from the pivot, and—most overlooked—the angle between actuator and arm throughout the entire stroke. This calculator is built for those realities. You’ll find the right torque formula, useful diagrams, and examples based on what actually trips up most installs.

What Is Torque from Force and Radius?

Torque is the rotational effect from a linear force, produced when a linear actuator pushes or pulls on an arm at a distance from a pivot point. But only the force component perpendicular to the lever arm generates usable torque. That depends on the changing angle between actuator and arm as the system moves.

How Does Actuator Force Become Torque?

If you think about using a wrench, all your pushing effort does nothing if you push along the handle’s length—you need to apply force sideways. Linear actuators work the same way on a hinging arm. Only the sideways (perpendicular) part of the actuator force creates torque, and that contribution varies with angle. At 90°, every pound counts. At shallower angles, most of the effort just fights the pivot—wasted as far as torque is concerned.

Engineering principle: Get your geometry right before you even think about force. Always check the minimum angle (where sin(θ) is lowest)—that’s where systems typically fail, not at a comfortable mid-stroke position.

"In nearly every failed actuator installation I've reviewed, the actuator was strong enough — the geometry wasn't. People size the actuator at the midpoint where the angle looks healthy, then discover the system stalls at the start of travel where the angle collapses to 15° or 20°. Check the sine of the angle at every position in the stroke before you pick a force rating." — Robbie Dickson, FIRGELLI Automations founder and former Rolls-Royce, BMW, and Ford engineer

Torque from Force and Radius Calculator

🎥 Video — Torque from Force and Radius Calculator

How Do You Use This Calculator?

It only takes a few steps to get an answer that matches the real world:

1. Pick your calculation mode. Use "Solve for Torque" if you have actuator force and need to know generated torque. "Solve for Required Actuator Force" if you have a torque requirement and need to choose an actuator.
2. Enter lever arm length. Measure from pivot/hinge center to where the actuator attaches to the arm. Always use the real mounting point, not just “somewhere near the end.” Enter in inches.
3. Enter the angle. This is between actuator and arm at the position you want to check. If you're uncertain, get a quick sketch or measure with a protractor—especially at the load peak.
4. Enter actuator force or target torque. Torque mode: actuator force in pounds. Force mode: desired torque in lb-ft.
5. Calculate the result. Then repeat for other key positions, especially where the angle is smallest, so you catch worst-case requirements.

What Is the Torque from Force and Radius Formula?

This is the practical formula you’ll use when sizing or checking actuator setups. Don’t skip the sine—every error we see comes from ignoring that.

Angle vs Mechanical Advantage Reference Table

Short version: If your actuator makes a 15° angle, you need nearly four times the force for the same torque as a 90° angle. Once you’re below 20°, focus on geometry—not just brute force actuator upgrades.

What Does a Simple Worked Example Look Like?

How Does This Apply to Real Engineering Problems?

The Angle Is Everything

If there’s one variable that rules actuator-driven hinge setups, it’s the angle between actuator and arm. At 90°, every bit of actuator force creates torque. At 45°, you’re down to 70%. At 30°, just 50%. Go to 15° and you’re only using 26%—so you need about four times more force than if you had ideal geometry. Small changes in angle make a huge difference. Don’t ignore this: it’s the main reason systems fail.

Torque Changes Through the Stroke

The angle between actuator and arm doesn’t stay constant as the arm swings—it changes continuously. That means torque drops and rises as you move. A heavy lid, for example, is hardest to open at the start (when the actuator is almost parallel to the arm), easy in the middle, and difficult again at the end—matching the swing of the actuator angle. Calculating just one position nearly always misses the problem.

Design for the Worst-Case Position

Most systems fail at the start or end of travel where the actuator angle is smallest and torque disadvantage is highest. Don’t rely on numbers from a “comfortable” mid-stroke point. Always size for the minimum angle encountered in the working range.

A Real-World Example: Opening a Heavy Lid

Let’s take a 60 lb lid. At the closed position, the actuator forms a 15° angle with the lid: sin(15°) = 0.259, so only about 26% of actuator force is useful. Need 200 lb-in to get started? With a 10" arm, you’ll need 200 ÷ (10 × 0.259) ≈ 77 lbs of actuator force at startup. By the time the angle hits 90°, that drops to about 20 lbs. This shows how much the starting angle increases your real requirements.

Mounting Rules of Thumb

Plan your actuator mounting to get as close to 90° at the hardest part of travel as you can. If the angle gets under 20° anywhere in the stroke, rethink your setup. When you drop below 20°, efficiency tanks and force requirements jump way up. Most actuator installs that stall or struggle missed this basic geometry check—especially at the travel endpoints.

How Do You Size an Actuator for a Variable-Angle Load?

What Are Common Mistakes When Using This Calculator?

1. Only calculating at the midpoint. That number is rarely your limiting case. Problems almost always start at minimum angle—usually at the stroke endpoints.
2. Using static force rating instead of dynamic. The force needed to hold is different than the force needed to move. Always source and use the actuator’s moving (dynamic) force.
3. Forgetting the sine. Near 0°, torque drops off steeply. Halving the angle can double the force required—don’t treat it as linear.
4. Leaving sub-20° angles in the design. At that point, the actuator is working extremely inefficiently. Adjust the mount instead of just oversizing.
5. No allowance for friction or load spikes. Real systems aren’t perfectly efficient. Add at least 20–30% margin for losses.
6. Measuring radius from the wrong spot. Always take the lever arm from pivot to where the actuator attaches to the arm—forget the other dimensions.

How Can You Verify the Calculator Output Is Reasonable?

1. Check at 90°. With θ = 90°, torque should be F × r exactly (since sin 90° = 1).
2. Check near 0° or 180°. At extreme angles, torque should approach zero.
3. At 30°, torque must be half the 90° value. This is a good sanity check.
4. Unit conversions should match. lb-in, lb-ft, and Nm outputs must all agree via standard factors.
5. Switching calculator modes should match your input. Calculated force should map back to original torque by re-running with the same inputs.
6. If you double force or arm length, torque must double. The relationship is linear with those two variables.
7. 1 lb at 1 ft at 90° = 1 lb-ft torque. If not, something’s wrong.

Frequently Asked Questions

Actuator issues nearly always come from ignored geometry, not actuator quality. Check the angle at every key position through the full travel, size for the worst case, and build in margin. If you need extra help matching actuator specs to your real torque requirements, browse our resources or reach out directly—we’ll help you make the numbers work on the bench, not just in the spreadsheet.

What Related Calculators Should You Use Next?

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