as far as the box does; so that the ball occupies only twothirds as much space as the cylinder which just surrounds it. Now imagine the ball moulded into a cone, with the radius of the ball as its altitude, and with the surface of the ball flattened into a circle for its base. The cone will be only one-half as tall as the box, for the box has the diameter of the ball for its altitude, and the cone has the radius of the ball for its altitude. Since the cone is one-half as tall as the box, to occupy as much space as the box does it would need a base six times as large as the base of the box (see 668); but since it occupies only twothirds as much space as the box does, its base need be only two-thirds of six times as large as the base of the box, or four times the base of the box. This experiment illustrates the truth, which can be proved by the principles of Geometry, that the surface of a sphere is four times the base of a cylinder in which the sphere is inscribed. 670. If you should cut the ball in halves, and then replace the halves in the box, the flat side of one half would exactly fit the bottom of the box, and the flat side of the other half would fit the top of the box, while the curved portions would just touch each other in the centre of the box. What portion of the box is empty? If you cut an orange in halves, the two circles forming the plane sides of the pieces contain as much surface as the peel on one of the pieces. Why? 671. If you should cut the box used in 669 and 670 along a line perpendicular to the two bases, and flatten the lateral surface into a plane surface, you would form a rectangle with the circumference of the original base as a base, and with an altitude equal to twice the radius of the base. If you should also mould one of the bases into a triangle, the triangle would have the circumference of the original base as a base, and an altitude equal to the radius. Since the rectangle has the same base as the triangle, but twice its altitude, the surface of the rectangle is four times that of the triangle. Make this clear by drawing a rectangle and a triangle which fulfil the above conditions, and by changing the triangle into a rectangle. Since the lateral surface of the cylindrical box is four times that of its base, it must be equivalent to that of the ball. Hence another way of describing the surface of a sphere: The surface of a sphere is the same as the lateral surface of a cylinder in which the sphere is inscribed. 672. Compare the entire surface of a cylinder with that. of a sphere which is inscribed in the cylinder. 673. Can you draw a circle that shall have as much surface as a given sphere has? A sphere is given if its radius is known. 674. Can you draw a cone that shall have a base whose radius is the radius of a given sphere, and that at the same time shall equal the sphere in volume ? 675. Find by arithmetic the volume of each of the following spheres, the radii of which are 7 in., 6 in., 3 in., and 14 in., respectively. Compare the first answer with the last answer, and the second answer with the third an swer. 676. Assuming that the earth is a sphere with a radius of 4,000 miles, how many square miles are there on the earth's surface? 677. As a result of your study of this section write and record a rule for finding the volume of each of the following solids: a parallelopiped; a prism; a pyramid; a cone; a cylinder; and a sphere. 678. Draw three lines, a, b, and c, and form a parallelopiped, a prism, a triangular pyramid, and a quadrangular pyramid, equivalent to the product of these lines. Can you draw a cone, a cylinder, and a sphere that shall be equivalent to the product of these lines? |