That is, the area of the surface of a sphere is equal four times the area of one of its great circles.
Exercises.
1. The convex surface of a cone is equal to half the rectangle contained by the circumference of the base and the slant height.
2. The convex surface of a right cylinder is equal to the rectangle contained by the circumference of the base and the altitude.
3. If P be a point in the base ABC of a triangular pyramid O–ABC, and if parallels to the edges OA, OB, OC, through P, meet the faces in the points a, b, c, the sum of the ratios
4. The volume of the frustum of a cone, made by a plane parallel to the base, is equal to the sum of the three cones whose bases are the two ends of the frustum, and the circle whose diameter is a mean proportional between the end diameters, and whose common altitude is equal to one-third of the altitude of the frustum.
5. If a point P be joined to the angular points A, B, C, D of a tetrahedron, and the joining lines, produced if necessary, meet the opposite faces in a, b, c, d, the sum of the ratios
6. The surface of a sphere is equal to the rectangle by its diameter, and the circumference of a great circle.