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 are vertical lines). Corresponding sides of these triangles are in the same ratio, so: If point "S" has coordinates ("x","m") then |"PS"| = |"y" - "m"| and the distance from "P" to the line is: Since "S" is on the line, we can find the value of m, and finally obtain: A variation of this proof is to place V at P and compute the area of the triangle ∆"UVT" two ways to obtain that formula_17 where D is the altitude of ∆"UVT" drawn to the hypoteneuse of ∆"UVT" from "P". The distance formula can then used to express formula_18,