Linkage Calculator
This linkage calculator finds the position of the slider in a slider-crank mechanism, the linkage that turns rotary motion into straight-line motion in engines, pumps and presses. A crank of fixed radius rotates about a center, and a connecting rod links the crank pin to a slider constrained to move along a line through that center. As the crank turns, the slider moves back and forth, and its distance from the crank center comes from the geometry: the projection of the crank along the slider line plus the projection of the connecting rod. The slider position equals the crank radius times the cosine of the crank angle plus the square root of the rod length squared minus the crank radius times the sine of the crank angle, squared. This is the standard kinematic relation used in automotive engine design, including the powertrain analysis behind vehicle safety work at the US National Highway Traffic Safety Administration. Enter the crank radius, the connecting-rod length and the crank angle, and the calculator returns the slider position. Every figure is computed deterministically from the slider-crank formula shown in full below, with a worked example that reconciles exactly to the calculator.
The slider-crank geometry gives the piston position: a crank radius of 50 mm, connecting rod 200 mm and crank angle 60 degrees place the slider 220.26 mm from the crank center. The slider reaches its farthest point at top dead center.
Slider-crank position formula
x = r cos(theta) + sqrt( L^2 - ( r sin(theta) )^2 )
r = crank radius, L = connecting-rod length
theta = crank angle from top dead center
x = slider distance from the crank center
The crank pin sits at r cos(theta) along the slider line and r sin(theta) to the side. The connecting rod spans from the pin to the slider, so its projection along the line is the square root of the rod length squared minus the sideways offset squared. Adding the two projections gives the slider position.
Worked example
Find the slider position for a crank radius of 50 mm, connecting rod of 200 mm and crank angle of 60 degrees.
- crank projection = r cos(theta) = 50 x cos(60 deg) = 50 x 0.5 = 25.00 mm
- sideways offset = r sin(theta) = 50 x sin(60 deg) = 50 x 0.866025 = 43.30 mm
- rod projection = sqrt(200^2 - 43.30^2) = sqrt(40,000 - 1,875) = sqrt(38,125) = 195.26 mm
- x = 25.00 + 195.26 = 220.26 mm
The slider sits 220.26 mm from the crank center. These are the calculator's default inputs, so the result above matches the widget exactly.
Slider position through one crank revolution
With a 50 mm crank and 200 mm rod, the slider sweeps between top and bottom dead center.
| Crank angle (deg) | Slider position (mm) |
|---|---|
| 0 | 250.00 |
| 60 | 220.26 |
| 90 | 193.65 |
| 120 | 170.26 |
| 180 | 150.00 |
Engine powertrain and transportation safety context: US National Highway Traffic Safety Administration (NHTSA).
Linkage Calculator: frequently asked questions
What is a slider-crank mechanism?
It is a four-bar linkage in which one link is replaced by a sliding joint, converting the rotation of a crank into the straight-line motion of a slider through a connecting rod. It is the heart of every reciprocating engine, where the piston is the slider, and of many pumps and presses. The position relation here gives the slider's distance from the crank center for any crank angle.
Where is top dead center?
Top dead center is the crank angle of zero, where the crank and connecting rod are in line and the slider is at its farthest point from the crank center, equal to the crank radius plus the rod length. Bottom dead center is at 180 degrees, where the slider is closest, at the rod length minus the crank radius. The stroke is twice the crank radius.
Why is the motion not a pure sine wave?
If the connecting rod were infinitely long the slider would move as a pure cosine of the crank angle. Because the rod has finite length, the rod-projection term adds a correction that makes the motion asymmetric: the slider spends slightly different times on each half of the stroke. This obliquity effect is captured by the square-root term.
What units does the calculator use?
It uses millimeters for the crank radius, connecting-rod length and slider position, and degrees for the crank angle. Any consistent length unit works as long as the crank radius and rod length share it; the slider position comes out in that same unit.
What if the crank radius exceeds the rod length?
For a working slider-crank the connecting rod must be longer than the crank radius, otherwise the rod cannot reach the slider line at some angles and the square-root term becomes imaginary. Keep the rod length greater than the crank radius so the geometry stays valid through a full revolution.
Official sources
- Vehicle powertrain and mechanical safety reference: US National Highway Traffic Safety Administration (NHTSA). As at 25 June 2026.
Reviewed by the CalculatorHub team, edited by James Graham, 25 June 2026. See our methodology. This is general information, not financial, tax, legal or investment advice.