Maps for Math

How does a crank turn rotation into a piston's back-and-forth motion?

The crank pin moves in a circle, and the connecting rod pushes the piston along a straight line. Trigonometry and Pythagoras give the piston's position: x = r cos θ + √(l² − r² sin² θ). A finite rod makes the piston reach half stroke before 90°, so it lingers near the bottom.

crankrodtop82.2 mmdashed: pure sine (endless rod)
Piston down from the top
82.2 mm
Piston speed at this angle
6.94 m/s
Peak piston speed
21.5 m/s
Mean piston speed
12.9 m/s
Crank angle at half stroke
79.7°

Challenge: Get the half-stroke angle within 1° of 90° using a rod no longer than 400 mm.

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Play

Slide the crank angle around a full turn. Then switch between the short-rod and long-rod presets.

Challenge: Get the half-stroke angle within 1° of 90° using a rod no longer than 400 mm. The box under the picture turns green when you get it.

Stuck? Pick one of the examples from the “Try an example” menu, or press “New example.”

Understand

s=r+l−(rcos⁡θ+l2−r2sin⁡2θ)s = r + l - \left(r\cos\theta + \sqrt{l^2 - r^2\sin^2\theta}\right)

The crank pin rides on a circle: (rcos⁡θ, rsin⁡θ)(r\cos\theta,\ r\sin\theta). The rod is the hypotenuse of a right triangle, so Pythagoras gives where it meets the piston:

x=rcos⁡θ+l2−r2sin⁡2θx = r\cos\theta + \sqrt{l^2 - r^2\sin^2\theta}

The graph compares the piston's travel with a pure sine wave (dashed). The shorter the rod, the more it bends away. The piston hurries through the top half and lingers in the bottom.

Use

Every input has a unit menu, so you can type values in the units you already have. Results follow your units.

Show the work

  1. Crank pin: a point on a circle of radius r (half the stroke)(r\cos\theta,\ r\sin\theta),\quad r = 43\ \text{mm}
  2. The rod is the hypotenuse: Pythagoras finds how far along the wrist pin sitsx = r\cos\theta + \sqrt{l^2 - r^2\sin^2\theta}
  3. Distance down from the top (x is largest at the top, r + l)s = 43 + 120 - \left(43\cos 150^\circ + \sqrt{120^2 - 43^2\sin^2 150^\circ}\right) = 82.18\ \text{mm}
  4. Mean piston speed: two strokes per turn\bar v = 2 \times 0.086\ \text{m} \times \frac{4500}{60} = 12.9\ \text{m/s}

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Turn the crank angle, then change the stroke and rod length. "More settings" sets the speed.

  • This is geometry only. Forces, balance, and combustion are left out.
  • The drawing is to scale, with the cylinder lying on its side.

For learning and estimation. Verify with applicable codes, standards, and a qualified professional before using in design, construction, or safety-critical work.

Cheat card

x=rcos⁡θ+l2−r2sin⁡2θx = r\cos\theta + \sqrt{l^2 - r^2\sin^2\theta}
s=r+l−xs = r + l - x
vˉ=2⋅stroke⋅n60\bar v = 2 \cdot \text{stroke} \cdot \frac{n}{60}
SymbolMeaningUnit
rrcrank radius (half the stroke)mm
llconnecting rod lengthmm
θ\thetacrank angle from top°
sspiston distance down from the topmm
  • With an endless rod, s = r(1 − cos θ), a pure sine wave.
  • The rod ratio l/r sets how far from a sine it is. Engines are usually about 3 to 4.
  • Peak piston speed comes before 90°, not at it.

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Where it’s used

  • Mechanical
    Engine designers pick the rod ratio to balance engine height, vibration, and piston side load.
  • Physics
    The crank-slider is the classic example of rotary motion becoming reciprocating motion, and back again.
  • Driving & Travel
    Every piston in a car engine follows this curve thousands of times a minute.

Questions people ask

What is the rod ratio?

The connecting rod length divided by the crank radius. A higher ratio makes the piston move more like a pure sine wave and pushes less on the cylinder wall.

What is mean piston speed?

The average speed of the piston over a turn, 2 × stroke × revolutions per second. Most production car engines stay under about 25 m/s.

Why doesn't the piston reach halfway at 90°?

At 90° the rod is tilted, which pulls the piston a little farther down than a sine wave would. So it passes half stroke before 90° and spends longer in the bottom half.

Sample TRG-0004 · rev 1 · model TRG-M04 · Level 4 · advanced algebra · updated 2026-09-27