Area of a circle "ds" over the whole circle formula_33 is just the arc length, which is its circumference, so this shows that the area "A" enclosed by the circle is equal to formula_34 times the circumference of the circle. Note that the area of a semicircle of radius r can be computed by the integral formula_35. By trigonometric substitution, we substitute formula_36, hence formula_37 The last step follows since the trigonometric identity formula_42 implies that formula_43 and formula_44 have equal integrals over the interval formula_45, using integration by substitution. But on the other hand, since formula_46, the sum of the two integrals is
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,