The document discusses the relationship between pi (π) and the circumference of circles. It states that the circumference of any circle equals π multiplied by its radius squared. It then explains that a cycloid curve, traced by a fixed point on a rolling circle, is close to but not quite circular, forming an elliptic segment. While the rolling circle moves at a fixed speed, the tracing point accelerates and decelerates, with its length equaling 8 times the radius and its area equaling 3 times pi multiplied by the radius squared.

1. 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 π r 2 (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 = 8 r ( 4 × the defining circle’s diameter (not pi dependent!)), area = 3 π r 2 ( 3 × the defining circle’s area).

2. π ZZA π ( Pi zza Pi e, Eight Radial Slices!) Visualizing the circumference (of any circle) as 3.14159… circle diameters (radius r × 2). Cycloid length = 8 r , area = 3 π r 2 0 1 2 3 4 Press “ Page Up ” to reset the animation, then Press “ Page Down ” to replay the animation Copyright © 2010, Henry Norman Relative Point Speed 0 10 5 Travelled distance = π = 3.14… circle diameters Travelled distance = 8 radii ( 4 circle diameters)