A lever sector uses a circular working edge centered on its pivot. When a contact or ideally non-slipping flexible element follows that edge, angular sweep corresponds to arc travel s = rθ.
Lever Sector Interactive Calculator
Vary sector radius and selected sweep angle to compare circular arc length, endpoint chord and travel per degree. The animation preserves a fixed radius throughout the sweep.
Equation Used
- The working edge is a circular arc exactly concentric with the fixed pivot.
- The selected angle is the total sweep between the two displayed endpoints.
- Any cable interpretation assumes ideal following without slip and is not a cable-drive design check.
Arc length, endpoint chord and travel per degree describe the circular geometry. Force and torque depend on the applied load and its direction; they are not calculated here.
Constant-radius sector geometry
Every point on the circular working edge is the same distance r from the pivot. Rotating through θ radians traces an arc of length s = rθ. The straight line joining the two endpoint positions is the chord c = 2r sin(θ/2).
The animation keeps the radius fixed while the arm sweeps from one endpoint to the other. It highlights only the portion of the rim already traversed, so the displayed progress cannot be mistaken for an independently changing radius.
Use the result as a geometry check
The calculator can check angular-to-linear travel on an ideal circular sector and compare that travel with the endpoint chord. It does not select a valve actuator, steering component, throttle quadrant, cable, chain or sector gear.
A physical design also needs its complete force path, contact and groove geometry, wrap, tension, friction, stiffness, bearings, stops, material strength, wear and applicable product requirements.
Arc and chord equations
The distance per degree is rπ/180. The endpoint chord is:
The chord is not cable take-up; it is included to distinguish travel along the rim from straight-line endpoint separation.
Worked geometry example
At r = 100 mm and θ = 30°, the angle is π/6 = 0.5236 rad. The arc length is 100×0.5236 = 52.36 mm. The endpoint chord is 2×100×sin(15°) = 51.76 mm, and travel per degree is 1.745 mm.
Model boundaries
A constant geometric radius does not guarantee constant input force, output torque or cable tension. Those depend on how forces are applied and on the rest of the mechanism. The calculation also omits slip, cable stretch and bending limits, groove shape, edge loading, bearing friction, backlash, deformation and strength.
The rendered arm and sector thickness are explanatory choices, not manufacturing dimensions.
Lever-sector questions
Why must the angle be in radians?
Radians are defined by the ratio of arc length to radius, which makes s = rθ direct. The calculator converts entered degrees before multiplying.
Is the chord the same as rim travel?
No. The chord is the straight line between endpoints; rim travel follows the circular arc and is longer except at zero sweep.
Does a circular sector guarantee constant torque?
No. A fixed radius is only one part of the force geometry. Torque also depends on force direction and magnitude.
Can this size a cable sector?
No. Cable diameter, groove, wrap, traction, bending fatigue, termination, tension and safety requirements are outside this model.
Geometry reference
- OpenStax, Algebra and Trigonometry, 7.1 Angles: defines radian measure and the circular arc-length relation s = rθ.
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