Math

Cycloid Calculator

Find the arc length and area of one cycloid arch.

Calculated Output

Arch length 16

Area under arch 37.699

Arch width 12.566

Formula: arc = 8r, area = 3πr², width = 2πr

Step-by-Step Breakdown

1. Values used:

2. Rolling circle radius = 2

4. Arch length = 16

5. Area under arch = 37.699

6. Rolling circle radius squared = 2 x 2 = 4

7. Arch width = pi x Rolling circle radius squared = 3.142 x 4 = 12.566

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The math behind it

One arch has length 8r, encloses area 3πr², and spans a width of 2πr (one full rotation).

Radius 2 → arch length 16, area ≈ 37.70.

Why is the area exactly 3 circles?

The area under one arch equals three times the rolling circle's area — a classic result of Galileo and Roberval.