Area of a circle Archimedes is not rigorous by modern standards, because it assumes that we can compare the length of arc of a circle to the length of a secant and a tangent line, and similar statements about the area, as geometrically evident. The area of a regular polygon is half its perimeter times the apothem. As the number of sides of the regular polygon increases, the polygon tends to a circle, and the apothem tends to the radius. This suggests that the area of a disk is half the circumference of its bounding circle times the radius. Following Archimedes' argument in "The
Instant centre of rotation and P-B. The direction of speed of point A is indicated by vector V. Its instant center of rotation must be perpendicular to this vector (as V is tangentially located on the circumference of a circle). The only line that fills the requirement is a line colinear with link P-A. Somewhere on this line there is a point P, the instant center of rotation for the body BAC. What applies to point A also applies to point B, therefore this instant center of rotation P is located on a line perpendicular to vector V, a line colinear with link P-B.