F 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 G H A D RULE. B Triangular Pyramid. Square Pyramid. Pentangular Pyramid. 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. 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. 255. 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. 1. Find the volume of a pyramid whose altitude is 12 ft. and the area of the base 15 sq. 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. B From ART. 145 we have the following RULE. To find the convex surface of a cylinder, multiply the circumference of the base by the altitude. WRITTEN EXERCISES. 1. Find the convex surface of a cylinder whose altitude is 10 ft. and the radius of the base 4 ft. 2. Find the convex surface of a cylinder whose altitude is 50 ft. and the diameter of the base 6 ft. 3. Find the entire surface of a cylinder whose altitude is 60 ft. and the diameter of the base 5 ft. 4. Find the entire surface of a cylinder whose altitude is 30 ft. and the radius of the base 7 ft. 257. We have found by ART. 253 that the volume of a prism equals the area of the base multiplied by the altitude. This will be true when the prism has an infinite number of sides and coincides with the cylinder. Hence the RULE. To find the volume of a cylinder, multiply the area of the base by the altitude. WRITTEN EXERCISES. 1. Required the contents of a cylinder 80 ft. long and 4 ft. in diameter. 2. Find the contents of a cylinder whose altitude is 4 rd. and the radius of the base 14 in. 3. Required the contents of a log whose length is 15 ft. and the diameter of the base 8 ft. 258. The Cone. A Cone is a round body whose base is a circle, and whose convex surface tapers uniformly to a point called the vertex. The altitude of a cone is the perpendicular distance from the vertex to the base; as, CO. The slant height is the distance from the vertex to the circumference of the base; as, CA. 259. We have found by ART. 254 that the convex surface of a pyramid equals the perimeter of the base multiplied by one-half the slant height. This will be true when the pyramid has an infinite number of sides and coincides with the cone. Hence the B RULE. To find the convex surface of a cone, multiply the circumference of the base by one-half the slant height. WRITTEN EXERCISES. 1. Find the convex surface of a cone whose slant height is 36 in. and the radius of the base 8 in. 2. Find the convex surface of a cone whose slant height is 60 ft. and the diameter of the base 20 ft. 3. A tent in the form of a cone has a slant height of 18 ft. and a diameter of 30 ft.; how many square yards of muslin are required to make it? 4. Find the entire surface of a cone whose slant height is 66 ft. and the radius of the base 22 ft. 5. Find the entire surface of a cone whose altitude is 15 in. and the radius of the base 8 in. 260. We have found by ART. 255 that the volume of a pyramid equals the area of the base multiplied by one-third of the altitude. This will be true when the pyramid has an infinite number of sides and coincides with the cone. Hence the RULE. To find the volume of a cone, multiply the area of the base by one-third of the altitude. WRITTEN EXERCISES. 1. Find the volume of a cone whose altitude is 12 ft. and the radius of the base 4 ft. 2. Find the volume of a cone whose altitude is 63 ft. and the diameter of the base 20 ft. 3. What is the volume of a cone whose slant height is 17 in. and the diameter of the base 16 in.? 4. Find the volume of a cone whose altitude is 72 ft. and the radius of the base 20 ft. 261. The Frustum of a Pyramid or Cone. The Frustum of a Pyramid is the part of the pyramid which remains after cutting off the top by a plane parallel to the base.. 262. The Frustum of a Cone is the part of a cone which remains after cutting off the top by a plane parallel to the base. |