1 SECTION XIX.-SOLIDS. 166. GEOMETRICAL DEFINITIONS. 1. A SOLID is a figure having length, breadth and thickness. 2 2. A PRISM is a solid whose ends are equal polygons, and whose sides are parallelograms. 3. A prism whose sides are all squares is called a cube.* If its ends are triangles, it is called a triangular prism. If the ends are squares, it is called a square prism,* if pentagons, a pentagonal prism, &c. C D 4. A CYLINDER is a round column whose ends are equal circles. 5. A PYRAMID is a solid having a triangle, a square, or a polygon for its base; its sides being triangles whose vertices meet in a point at the top, called the vertex of the pyramid. 6. A CONE is a solid having a circular base, and tapering uniformly to a point at the top. * A solid contained by six quadrilateral planes, every opposite two of which are equal and parallel, is called a parallelopipedon. E F 7. A SEGMENT of a solid is the part cut off from the top by a plane parallel to its base. 8. A FRUSTUM is the part left at the bottom after the segment has been cut off. 9. The axis of a solid is a line drawn from the middle of one end to the middle of the opposite end. 12. The axis or diameter of a sphere is a line passing through the centre and terminating at the surface, as the line A B. 13. The height of a solid is the perpendicular distance between its top or vertex and its base. 14. The slant height of a pyramid is a line drawn from the vertex to the middle of one of the sides at the base. 15. The slant height of a cone is the shortest line that can be drawn from the vertex to the circumference of the base. 16. A spheroid is a solid generated by the revolution of an ellipse about one of its diameters. If the ellipse revolves about its longer diameter, the solid is called prolate or oblong spheroid; if about its shorter diameter, an oblate or flattened spheroid. 167. MENSURATION OF SOLIDS. TO FIND THE AREA OF THE SURFACE OF A CUBE. RULE. Multiply the square of the length of one side by 6; the prod uct will be the area. (Why?) 1. The side of a cube is 15 inches; what is the area of its surface? 2. What is the area of the surface of a cube, the side of which is 10 inches? 1 ft. 6 in. ? 5 ft. 8 in.? 168. TO FIND THE SOLID CONTENTS OF A CUBE. RULE. Cube the side given. 3. How many solid feet in a cubical block of marble, the length of which is 15 inches? 1 ft. 4 in.? 2 ft. 5 in.? 169. TO FIND THE SURFACE OF A PRISM, PARALLELOPIPEDON, OR CYLINDER. RULE. Multiply the perimeter or circumference of the base by the height, and to this product add the area of the two ends; the sum will be the area. 5. What is the surface of a triangular prism, whose length is 12 feet, and each of its equal sides 4 feet? 5 ft.? (157, Rule II.) 6. What is the surface of a square pyramid, or parallelopipedon, the height of which is 15 feet, and each side of its base 3 feet? 7. What is the convex surface of a cylinder 15 inches long, and its base 15 inches in circumference? 1 ft. 9 in. long, and 1 ft. in diameter ? 8. What is the whole surface of a cylinder, whose length is 25 feet, and the diameter of its base 4 feet? (3.1416 × 4×25) + (42 × 7854 × 2) = the answer. 9. Required the inside and the outside surface of a box, measuring 4 feet long, 2 feet wide, and 3 feet deep on the outside, the boards of which it is made being 1 inch thick. 170. TO FIND THE SOLID CONTENTS OF A PRISM OR CYLINDER. RULE. Multiply the area of the base by the altitude; the product will be the solidity. ft.? 10. What is the solidity of a triangular prism, whose length is 12 feet, and each of its equal sides 4 feet? 5 11. What are the solid contents of a square pyramid, the height of which is 15 feet, and each side of its base 3 feet? 12. What is the solidity of a cylinder 15 inches long, and its diameter 6 inches? 1 ft. 9 in. long, and 1 foot in diameter? 13. How many solid inches in a grindstone 28 inches in diameter, and 5 inches thick? 14. How many solid feet in a round cistern 5 feet in diameter, and 8 feet deep? 15. If a cubic foot contains 7 gallons, how many gallons will the above cistern hold? 171. TO FIND THE SURFACE OF A PYRAMID OR CONE. RULE. Multiply the perimeter, or the circumference of the base, by one half the slant height. 16. What is the surface of a pyramid, the perimeter of its base being 15 feet, and its slant height 25 feet? perimeter 12, and the slant height 20 inches? 17. What is the convex surface of a cone whose slant height is 20 inches, and the circumference of its base 15 inches? 18. What will be the expense of painting a conical spire of which the height is 118 feet, and the circumference of the base 64 feet, at 8d. per square yard? 172. TO FIND THE SOLID CONTENTS OF A PYRAMID OR CONE. RULE. Multiply the area of the base by one third of the altitude; the product will be its solidity. 19. What are the solid contents of a square pyramid, of which the height is 42 inches, and one side of the base 14 inches? if the height be 12 ft. 3 in., and a side of the base 1 ft. 4 in. ? 20. What is the solidity of a cone whose height is 15 ft., and the diameter of the base 2 ft. 4 inches? 21. How many solid ft. in a square stick of timber that tapers towards one end to a point, the length being 20 feet and a side of the base 25 inches ? 173. TO FIND THE SURFACE OF A FRUSTUM OF A PYRAMID OR CONE. RULE. Add the perimeters or circumferences of the two ends together, and multiply half the sum by the slant height, for the upright or curve surface, to which add the areas of the two ends, to get the whole surface. 22. Required the surface of a frustum of a square pyramid, the sides of the ends being 40 and 26 inches, and the slant height 10 feet. 23. Required the surface of the frustum of a cone, the diameters of the bases being 43 and 23 inches, and the slant height 9 feet. 24. What will be the cost of dress ing the upright surface of a square stone pillar, of which each side of the base is 3 feet, and of the top 2 feet, the pillar being 15 feet high, at 12 cents per square foot? 174. TO FIND THE SOLID CONTENT OF THE FRUSTUM OF A PYRAMID OR CONE. RULE. Find the areas of the two ends, and take the square root of their product. To this add the two areas; the sum, multiplied by one third of the perpendicular height, will give the solid content. |