cylinder which has the same base and height. Proposition 11: The volume of a cone (or cylinder) of the same height is proportional to the area of the base. Proposition 12: The volume of a cone (or cylinder) that is similar to another is proportional to the cube of the ratio of the diameters of the bases. Proposition 18: The volume of a sphere is proportional to the cube of its diameter. Archimedes Archimedes used the method of exhaustion as a way to compute the area inside a circle by filling the circle with a polygon of a greater area and greater number of sides. The quotient formed by the area of this polygon divided by the square of the circle radius can be made arbitrarily close to π as the number of polygon sides becomes large, proving that the area inside the circle of radius r is πr2, π being defined as the ratio of the circumference to the diameter (C/d). He also provided the bounds 3 + 10/71 < π < 3 + 10/70, (giving a range of 1/497) by comparing the perimeters of the circle with the perimeters of the inscribed and circumscribed 96-sided regular polygons. Other results he obtained with the method of exhaustion included The area bounded by the intersection of a line and a parabola is 4/3 that of the triangle having the same base and height(the quadrature of the parabola); The area of an ellipse is proportional to a rectangle having sides equal to its major and minor axes; The volume of a sphere is 4 times that of a cone having a base of the same radius and height equal to this radius; The volume of a cylinder having a height equal to its diameter is 3/2 that of a sphere having the same diameter; The area bounded by one spiral rotation and a line is 1/3 that of the circle having a radius equal to the line segment length; Use of the method of exhaustion also led to the successful evaluation of an infinite geometric series (for the first time); See also The Method of Mechanical Theorems The Quadrature of the Parabola Trapezoidal rule Pythagorean Theorem References Volume Euclidean geometry Integral calculus History of mathematics 5th century BC in Greece