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
Director circle ellipse. The Apollonius circle, the locus of points "X" such that the ratio of distances of "X" to two foci "P" and "P" is a fixed constant "r", is another special case, with "w" = 1, "w" = −"r", and "C" = 0. In the case of a parabola the director circle degenerates to a straight line, the directrix of the parabola. Director circle In geometry, the director circle of an ellipse or hyperbola (also called the orthoptic circle or Fermat–Apollonius circle) is a circle consisting of all points where two perpendicular tangent lines to the ellipse or hyperbola cross