The next slide is a graphic visualization of the fact that Pi (π) is always 3.14159… diameters, of any given circle, which is why the circumference, of any circle, equals πr2 (where r is the radius of the given circle).

The cycloid curve—traced by a fixed point on a circle rolling smoothly in a straight line (e.g., a railway wagon wheel)—is close to but not quite circular: it forms an elliptic segment.

While the circle is rolled at a fixed speed, the cycloid tracing point moves at varying speeds: at rest, accelerating to the top of the arch, then decelerating back to rest, and so on.

Cycloid length = 8r (4× the defining circle’s diameter (not pi dependent!)), area = 3πr2

(3× the defining circle’s area).

Travelled distance = π = 3.14… circle diameters

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πZZAπ (Pizza Pie, Eight Radial Slices!)

Visualizing the circumference (of any circle)as 3.14159… circle diameters (radius r × 2).

Cycloid length = 8r, area = 3πr2

Press “Page Up” to reset the animation, thenPress “Page Down” to replay the animation

Copyright © 2010, Henry Norman

RelativePoint Speed

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Travelled distance = 8 radii (4 circle diameters)