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
Distance from a point to a line formula_19, and formula_20in terms of the coordinates of P and the coefficients of the equation of the line to get the indicated formula. Let "P" be the point with coordinates ("x", "y") and let the given line have equation "ax" + "by" + "c" = 0. Also, let "Q" = ("x", "y") be any point on this line and n the vector ("a", "b") starting at point "Q". The vector n is perpendicular to the line, and the distance "d" from point "P" to the line is equal to the length of the orthogonal projection of formula_21 on n. The