SOLIDS. 251. A Solid is that which has length, breadth, and thickness. The Volume of a solid is the number of times that it contains another solid regarded as the unit of measure. 252. The Prism. A Prism is a solid whose ends are equal polygons and whose sides are parallelograms. The bases are the equal polygons, and the parallelograms form the convex surface. Prisms are named from their bases. A Rectangular Parallelopiped, or Square Prism, is a prism whose six faces are all rectangles. A Cube is a rectangular parallelopiped whose six faces are all squares. A Right Prism is a prism whose edges are perpendicular to its bases: the prisms in the cut above are right prisms. RULE. The convex surface of a right prism is equal to the perimeter of the base multiplied by the altitude. WRITTEN EXERCISES. 1. Find the convex surface of a triangular prism the three sides of whose base are 6, 8, and 10 in., respectively, and altitude 40 in. 2. Find the convex surface of a square prism the sides of whose base are each 10 in. and altitude 50 in. 3. Find the convex surface of a pentangular prism, each of the sides of the base being 8 ft. and the altitude 12 ft. 4. Find the surface of a cube whose edge is 6 inches. 5. Find the entire surface of a triangular prism the sides of whose base are 8, 10, and 12 in., respectively, and altitude 20 in. NOTE.—To find the entire surface, add the area of the bases. 6. Find the entire surface of a triangular prism the sides of whose base are each 20 ft. and altitude 40 ft. E D 253. Let the figure ABCDEF represent a rectangular parallelopiped, or square prism, whose length is 4 in., width 3 in., and altitude 9 in.; what is its volume? If we divide the length, width, and height into inches and pass planes through these points of division, the prism will be divided into equal cubes. There will be 12 cubic inches in one layer; and since there are 9 such layers, the volume will equal 9 times 12 cubic inches, or 108 cubic inches. Hence the B RULE. The volume of a rectangular prism is equal to the area of the base multiplied by the altitude. A rectangular prism may be divided into two equal triangular prisms, each having the altitude of the rectangular prism and a base equal to half the base of the rectangular prism. Hence the RULE. The volume of a triangular prism is equal to the area of the base multiplied by the altitude. Any prism may be divided into triangular prisms, each of which is equal to the area of the base multiplied by the common altitude, and their sum, which is the volume of the prism, is equal to the sum of their bases or the base of the prism multiplied by the altitude. Hence the general RULE. The volume of any prism is equal to the area of the base multiplied by the altitude. WRITTEN EXERCISES. 1. Find the volume of a square prism whose altitude is 40 ft. and the sides of the base 4 ft. 2. What is the volume of a rectangular prism whose altitude is 60 ft. and the sides of the base 4 ft. and 8 ft., respectively? 3. Find the volume of a triangular prism whose altitude is 100 ft. and the sides of the base each 20 ft. 4. Find the volume of a triangular prism whose altitude is 40 ft. and the base a right triangle, with the sides about the right angle 8 ft. and 12 ft., respectively. 5. The water in a stream flows at the rate of 3 miles an hour; if the stream is 4 ft. wide and 3 ft. deep, how many cubic yards of water pass a given point in 24 hours ? 254. The Pyramid. A Pyramid is a solid bounded by a polygon and by several triangles meeting in a point; as, ABCD - E. The polygon is called the base of the pyramid, and the triangles form the convex surface. The altitude is the perpendic ular distance EO from the vertex to the base. The slant height of a pyramid is the altitude of any one of its lateral faces; as, EN Pyramids are named from their bases. RULE. The convex surface of a right pyramid is equal to the perimeter of the base multiplied by half the slant height. WRITTEN EXERCISES. 1. Find the convex surface of a square pyramid whose slant height is 24 ft. and the sides of the base each 5 ft. 2. Find the convex surface of a triangular pyramid whose slant height is 60 ft. and the sides of the base each 21 ft. 3. Find the convex surface of a pyramid whose slant height is 48 ft. and the base a hexagon whose sides are each 16 ft. 4. Find the entire surface of a square pyramid whose slant height is 50 ft. and each side of the base 20 ft. 5. Find the entire surface of a triangular pyramid whose slant height is 36 in. and the sides of the base are each 8 in. 257. If we make two vessels of tin, the one a pyramid and the other a prism, having equal bases and equal altitudes, the prism will hold just three times as much as the pyramid. Hence the RULE. To find the volume of a pyramid, multiply the area of the base by one-third the altitude. WRITTEN EXERCISES. sq. 1. Find the volume of a pyramid whose altitude is 12 ft. and the area of the base 15 ft. 2. Find the volume of a pyramid whose altitude is 45 ft. and the base a square 10 ft. on a side. 3. What is the volume of a rectangular pyramid whose altitude is 60 ft. and the sides of the base 12 and 20 ft., respectively? 4. Find the volume of a pyramid whose altitude is 72 ft. and the base a triangle 24 ft. on each side. 5. Find the volume of a pyramid whose altitude is 24 ft., and having a right triangle for its base whose sides are 8, 15, and 17 ft., respectively. 256. The Cylinder. A Cylinder is a round body with equal and parallel circles for its bases and having a uniform diameter. The altitude of a cylinder is the perpendicular distance between its bases. The convex surface of a cylinder is the curved surface which bounds it. Cylinder. |