Sphere

Sphere is the most regular and symmetrical solid figure. It is produced by the revolution of a semicircle about its diameter, and every point on its surface is equidistant from its centre. every plane cuts the sphere in a circle; if the plane passes through the centre, the circle is caled a great circle, other circles being called small circles. All great circles are equal. Two spheres always intersect in a circle, whose plane is perpendicular to the line joining the centre of the spheres. The surface of a sphere is equal to 4 pi r², where r is its radius, and is equal to 2/3 the total surface of the circumscribing cylinder. This cylinder is one whose length and the diameter of whose ends are equal to the diameter of the sphere. Its center therefore coincides with the center of the sphere, and the latter is just contained in it. If we regard only the curved surface of the cylinder and not the ends, we note that the surface of sphere and cylinder are equal; also, if we take a section of the sphere parallel to the base of the cylinder, the curved surface of the portion of the sphere so cut off is equal to that of the cylinder. But the area of the curved surface of the cylinder equals the circumference of its base (which is the same as that of a great circle of the sphere) multiplied by its height; hence this is the area of the section of the sphere's surface. Extending this slightly, we see that if a sphere be cut by two parallel planes, the area of the curved surface so obtained is equal to the distance between two planes multiplied by the circumference of a great circle. The volume of a sphere is 4/3 pi r³, or 2/3 the volume of the circumscribing cylinder.