Free · Algebra and geometric relations

One Cycloid Arch Measures

Calculate the dimensions, arc length and area of one complete cycloid arch.

Formula & worked example
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How to use this calculator

Enter the known values in the units shown. Results update as you type.
Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
Use the worked example to check the method. Reset restores the starting fields.

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Formula and method

For the worked example, Arc Length (S) = 16, Hump Height (d) = 4.

Arc=8r; enclosed area=3πr²; horizontal span=2πr

Worked example

Enter these known values and leave the other values blank.

Radius of rolling circle (r): 2 m

Area of cycloid (A): 37.7 m²
Arc Length (S): 16 m
Hump Height (d): 4 m
Hump Length (C): 12.566 m
Perimeter of cycloid: 28.566 m

Assumptions and limitations

A circle rolls without slipping on a straight baseline for one full revolution.
Perimeter includes both the curved arch and its straight baseline.

Common questions

How do I use this relation?

Choose the relevant mode and enter independent known quantities. Leave a missing value empty where the control permits an inverse.

What do the displayed numbers mean?

The result follows the stated mathematical model. All dimensions must use consistent units; available unit selectors handle conversions.

References

The calculation equations, inverse formulas, units, and input rules were imported from this source. Bookify provides the interface and equation solver.

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