Enter the radius of the Ferris wheel, the period of rotation, and the elapsed time into the calculator to determine the height above ground of a point on the wheel. Optionally enter the axle (center) height above ground; if left blank, the calculator assumes the wheel’s lowest point is at ground level (axle height equals radius).

Ferris Wheel Equation Calculator

Enter Radius (R), Period (T), and Elapsed Time (t) to calculate Height (H). Axle Height (C) is optional.

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Ferris Wheel Equation

The following equation is used to calculate the height above ground of a point on a Ferris wheel at time t.

H(t)=C-R\cos\left(\frac{2\pi t}{T}\right)

Variables:

  • H(t) is the height above ground at time t (meters)
  • C is the axle (center) height above ground (meters)
  • R is the radius of the Ferris wheel (meters)
  • T is the period of rotation (seconds)
  • t is the elapsed time since the point was at the lowest position (seconds)

To calculate the height above ground at time t, compute the angle of rotation (2πt/T) and use H(t)=C−R·cos(2πt/T). If the wheel’s lowest point is at ground level, then C=R, and the height ranges from 0 to 2R.

What is the Ferris Wheel Equation?

The Ferris wheel equation models the vertical position of a point on the edge of a rotating Ferris wheel over time. It uses the wheel radius, the rotation period, and a vertical offset (the axle/center height above ground). With a chosen starting position (here, t = 0 at the lowest point), the height varies sinusoidally, which is a form of simple harmonic motion.

How to Calculate Height on a Ferris Wheel?

The following steps outline how to calculate the height above ground of a point on a Ferris wheel.

  1. Determine the radius of the Ferris wheel (R).
  2. Determine the period of rotation (T).
  3. Determine the elapsed time (t) since the point was at the lowest position.
  4. Determine the axle (center) height above ground (C). If the wheel’s lowest point is at ground level, you can use C = R.
  5. Use the formula: H(t)=C−R·cos(2πt/T).
  6. Calculate the height above ground (H) and (optionally) check your answer with the calculator above.

Example Problem : 

Use the following variables as an example problem to test your knowledge.

Radius of the Ferris wheel (R) = 10 meters

Period of rotation (T) = 60 seconds

Assume the wheel’s lowest point is at ground level, so the axle height is C = R = 10 meters.

Find the height after t = 10 seconds:

H(10)=10−10·cos(2π·10/60)=10−10·cos(π/3)=10−10·(0.5)=5 meters.