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

You've mounted a linear actuator to a hinged lid, gate, or arm — and it won't budge. Or worse, it opens fine at the midpoint but stalls at the start. The problem isn't the actuator. It's the geometry. Torque depends on where the actuator connects, how far that point sits from the pivot, and — critically — the angle between the actuator and the arm at every position through the stroke. This calculator handles all of it. You'll get the torque formula, worked examples, a visual diagram, and the engineering insight to avoid the most common mounting mistake we see.

What Is Torque from Force and Radius?

Torque is the rotational force produced when a linear actuator pushes or pulls on an arm at some distance from a pivot. Only the component of force perpendicular to the arm creates torque — and that depends on the angle between the actuator and the arm.

How Does Actuator Force Become Torque?

Think of a wrench on a bolt. Push straight along the handle and nothing happens — you need to push sideways to turn it. A linear actuator on a hinged arm works the same way. The force only counts if it has a sideways component relative to the arm, and that component changes as the arm rotates. At 90° you get full force converted to torque. At shallow angles, most of the force just pushes uselessly into the pivot.

Engineering principle: Motion design starts with geometry, not force alone. Measure the hard part of travel — the angle where sin(θ) is smallest — not the comfortable middle.

"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?

Getting accurate results takes about 30 seconds. Here's how.

1. Choose your calculation mode. Select "Solve for Torque" if you know the actuator force and want to find the torque it produces. Select "Solve for Required Actuator Force" if you know the torque you need and want to find what force rating the actuator must have.
2. Enter the lever arm length. Measure from the center of the pivot or hinge to the exact point where the actuator rod connects to the arm. Use inches.
3. Enter the angle. This is the angle between the actuator's line of force and the lever arm at the specific position you're analyzing. If you're not sure, sketch it out — or measure it with a protractor at the position of maximum load.
4. Enter force or target torque. In torque mode, enter the actuator's rated force in pounds. In force mode, enter the torque you need in lb-ft.
5. Hit Calculate. Review the results, then re-run with different angles to understand how torque changes through the full range of motion.

What Is the Torque from Force and Radius Formula?

The core formula connects linear actuator force to rotational torque through the geometry of the installation.

Angle vs Mechanical Advantage Reference Table

Read this as: at 15°, you need nearly 4× the actuator force to produce the same torque you'd get at 90°. Below 20°, redesign the geometry rather than oversizing the actuator.

What Does a Simple Worked Example Look Like?

How Does This Apply to Real Engineering Problems?

The Angle Is Everything

The angle between the actuator and the arm is the single most important variable in any actuator-driven hinge application — and it's the most misunderstood. At 90° you convert 100% of the actuator's force into usable torque. Drop to 45° and you only get 70.7%. At 30° you're down to 50%. At 15° you're at just 25.9%, which means you need nearly 4 times the actuator force to produce the same torque you'd get at 90°. That's the difference between a smooth, reliable system and one that strains, stalls, or burns out the actuator.

Torque Changes Through the Stroke

Here's what catches most people off guard — as a hinged panel or arm rotates, the angle between the actuator and the arm changes continuously. Torque is not constant through the stroke. A heavy lid might start nearly closed with the actuator pushing at just 10° or 15° to the arm, then pass through 90° at the midpoint, then back to a shallow angle at full open. The torque curve follows the sine of that angle, so it peaks in the middle and drops off at both ends. This means you can't just calculate torque at one position and call it done.

Design for the Worst-Case Position

The worst case is almost always at the start or end of travel where the angle is smallest. If you're lifting a lid, the maximum load (fighting gravity) typically happens right when the actuator's angle to the arm is at its worst. You need to calculate the required force at that specific position — not the comfortable midpoint where the angle is larger and the math looks better. Design for the hard part, not the easy part.

A Real-World Example: Opening a Heavy Lid

Consider a storage box with a 60 lb lid hinged at the back. At the closed position, the actuator might only make a 15° angle with the lid. sin(15°) = 0.259 — so only 25.9% of the actuator's force actually produces torque. If you need 200 lb-in of torque to crack that lid open, you'd need 200 ÷ (r × 0.259) force from the actuator. With a 10-inch lever arm, that's about 77 lbs of actuator force — but at the 90° midpoint you'd only need about 20 lbs. That's a nearly 4x difference, and it's why undersized actuators fail at the start of travel.

Mounting Rules of Thumb

Try to mount the actuator so it's as close to 90° to the arm as possible at the point of maximum load. If the angle drops below 20° anywhere in the travel, seriously reconsider your geometry. Below 20° the mechanical disadvantage becomes severe — force requirements spike exponentially, and you're asking the actuator to work dramatically harder than its rating suggests. This is one of the most common mistakes we see in customer installations. The geometry at the mounting position wasn't checked through the full range of motion, and the actuator that looked perfect on paper can't handle the real-world angles.

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

What Are Common Mistakes When Using This Calculator?

1. Calculating at the midpoint only. The angle changes through the entire stroke. Running the math at θ = 90° tells you nothing about whether the actuator can break loose at θ = 15°. Always check the worst-case position in the travel.
2. Plugging in the static force rating instead of dynamic force. The static (holding) rating is higher than the dynamic (moving) rating. For starting motion under load, use the dynamic force the actuator can produce at stall.
3. Ignoring the sine relationship. Small angle changes near 0° produce huge force changes. Going from 30° to 15° doubles your required force; going from 15° to 10° increases it another 50%. This is non-linear.
4. Accepting angles below 20° in the design. Below 20°, mechanical disadvantage is severe. Redesign the geometry — move the base mount, add a bell crank, or relocate the attachment point — rather than oversizing the actuator.
5. Forgetting the friction and efficiency margin. The calculator returns ideal theoretical torque. Real hinges have friction, mounts have slop, and motion has dynamic loads. Add 20–30% to your calculated force requirement; more if the application has shock loads.
6. Measuring radius from the wrong point. The radius is the distance from the pivot to where the actuator rod attaches to the arm — not to the load, not to the end of the arm, and not to the actuator's base mount.

How Can You Verify the Calculator Output Is Reasonable?

1. Boundary check at 90°. Set θ = 90°. Torque should equal F × r exactly (sin 90° = 1.000). If you enter 100 lbs and 12 in at 90°, the result must be 1,200 lb-in = 100 lb-ft. If it isn't, the inputs are wrong.
2. Boundary check near 0° and 180°. As θ approaches 0° or 180°, torque should approach zero. At θ = 1° the output should be roughly 1.7% of the 90° value. At θ = 179° the same.
3. Halving check at 30°. sin(30°) = 0.500 exactly. Torque at 30° must equal half the torque at 90° for the same F and r.
4. Unit conversion cross-check. Take the lb-in result, divide by 12 — it must equal the lb-ft result. Multiply lb-in by 0.11298 — it must equal the Nm result. If the three units don't agree internally, something is off.
5. Round-trip between the two modes. Compute torque from force at known F, r, θ. Then switch to "Solve for Force", paste the torque back in with the same r and θ, and you should recover the original force value within rounding error.
6. Sanity-check against the symmetric arm rule. If you double the lever arm length, torque must double. If you double the force, torque must double. Linear relationships in both.
7. Compare against a known reference. A 1 lb force at 1 ft (12 in) at 90° must produce exactly 1 lb-ft of torque. This is the definition. If the calculator disagrees, treat it as a configuration error.

Frequently Asked Questions

The most common failure we see in actuator projects isn't a bad actuator — it's bad geometry. Run your numbers at every position through the full stroke, design for the worst-case angle, and add a safety margin. Do that, and you'll build something that works reliably for years. If you need help selecting the right actuator for your torque requirements, check our product range or reach out — we're happy to help you get the geometry right.

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