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 a side, and "h" is the perpendicular height from the opposite vertex. This proof is only valid if the line is neither vertical nor horizontal, that is, we assume that neither "a" nor "b" in the equation of the line is zero. The line with equation "ax" + "by" + "c" = 0 has slope -"a"/"b", so any line perpendicular to it will have slope "b"/"a" (the negative reciprocal). Let ("m", "n") be the point of intersection of the line "ax" + "by" + "c" = 0 and the line perpendicular to it which passes through the point ("x", "y").