MENSURATION. CHAPTER XVII. SOLIDS. AREAS, VOLUMES AND WEIGHTS OF PRISMS (RIGHT AND OBLIQUE). CYLINDER, HOLLOW CYLINDER, PYRAMID, CONE, FRUSTUM OF PYRAMID, FRUSTUM OF CONE. GULDIN'S THEOREM. SURFACE AND VOLUME OF SPHERE, CYLINDER, RING, FLY-WHEEL. A solid figure or solid is a figure having the three dimensions of length, breadth and thickness. When the surfaces bounding a solid are plane, they are called faces, and the edges of the solid are the lines of intersection of the planes forming its faces. What are called the regular solids are five in number, viz., the cube, tetrahedron, octahedron, dodecahedron, and icosahedron. The cube is a solid having six equal square faces. The tetrahedron has four faces all equilateral triangles. ไป The octahedron has eight faces all equilateral triangles. The dodecahedron has twelve faces all pentagons. The icosahedron has twenty faces all equilateral triangles. Models of these five solids can be made from cardboard by drawing the developments of the solids to any convenient size. Cylinder. If a rectangle ABCD (Fig. 140) be made to revolve about one side AB, as an axis, it will trace out a cylinder. Or, a cylinder is traced by a straight line always moving parallel to itself round the boundary of any curve, called the guiding curve. FIG. 140. Pyramid.-If one end of the line always passes through a fixed point, and the other end be made to move round the boundary of any curve, a pyramid is traced out. Cone. If the curve be a circle and the fixed point is in the line passing through the centre of the circle, and at right angles to its plane, a right cone is obtained; an oblique cone results when the fixed point is not in the line at right angles to the plane of the base. C FIG. 141. A FIG. 142. B Sphere. If a semi-circle ACB (Fig. 142) revolve about a diameter AB, the surface generated is a sphere. FIG. 143. Prism. When the line remains parallel to itself and is made to pass round the boundary of any rectilinear polygon, the solid formed is called a prism. The ends of a prism and the base of a pyramid may be polygons of any number of sides, i.e., triangular, rectangular, pentagonal, etc. A prism is called triangular, rectangular, square, pentagonal, hexagonal, etc., (Fig. 143) according as the end or base is one or other of these polygons. A prism which has six faces all parallelograms is called a parallelopiped. A right prism has its side faces perpendicular to its ends; other prisms are called oblique. F E D B FIG. 144.-Volume of a right prism or parallelopiped. In (Fig. 144) a right prism, the ends of which are rectangles, is shown; to find its volume it is necessary to find the area of one end DCGE, and multiply by the length BC. Let 1, b and d denote the length, breadth, and depth respectively. Then area of one end = bxd. As b xl area of base; volume = = area of base × altitude. Ex. 1. If the length be 8 ft., the depth 2 ft., and breadth 3 ft. Area of one end=3x2=6 sq. ft.; .. volume 6x8 48 cub. ft. If, as in Fig. 144, the length BC be divided into 8 equal parts, the breadth into 3, and the depth into 2. that there are 6 square units in the end hence the volume is 8 × 6=48 cubic feet. Then it will be seen Total Surface of a Right Prism.—The total surface is, from Fig. 144, seen to be twice the area of the face ABCD, and twice the area of ADEF, together with the area of the two ends. .. Surface 2(ld+bl+bd), or perimeter of base multiplied by altitude together with areas of the two ends. Volume of a Cube.-When the length, breadth, and depth are each equal the solid is called a cube, and if a denote the length of edge of the solid the area of the base is a2 and the volume is a3. Ex. 2. Let the length of the edge of the cube be 4 inches (Fig. 145). It is seen by dividing two adjacent sides (or edges), as EB and BC, each into 4 equal parts and drawing through the points of division lines parallel to EB and BC that one end BCDE contains 16 square inches, and as there are four such rows, the volume A is 4 x 16 64 cubic inches. E B FIG. 145.-Volume of a cube. Volume of an Oblique Prism.-The volumes of all parallelopipeds having the same or equal bases and the same altitudes are equal. In Fig. 146 an oblique prism ABCDEF is shown. By drawing CN and DH perpendicular to DC, and NP parallel to BF, wedge-shaped pieces are obtained. Assuming the wedgeshaped piece CNPFB transferred to the left, as indicated, the oblique prism becomes a right prism on a rectangular base. Thus the volume of an oblique prism or parallelopiped is equal to the area of the base multiplied by its altitude. Perhaps the best and the easiest method is to build up a rectangular prism from a number of thin rectangles (of millboard, cardboard, or thin wood). By shearing the solid so constructed, as in Fig. 147, any degree of obliquity can be obtained, and the height, which is obviously the sum of the thicknesses of the rectangles, remains the same. Hence the volume of the solid is unaltered. FIG. 147.-Model to illustrate a right and an oblique prism. When the volume of any solid is known, then the weight can be obtained by multiplying the volume by the weight of unit volume. The weights of unit volumes of the materials usually used are given in Table III., p. 371. Thus, a rectangular piece of wrought iron, of thickness inch, 8 inches wide, and 12 inches in length, would weigh 1. Find the volume of a beam 10 ft. 6 in. long, 4 in. wide, and 2 in. thick. 2. Find the solid content of a log of timber 10 yds. 2 ft. 7 in. long, 2 ft. 11 in. broad, and 2 ft. 5 in. thick. 3. The length of a ditch, horizontal at top and bottom, is 100 yds., its depth 12 ft., width at top 12 ft., and at bottom 8 ft. Find its content. 4. A ditch is 5000 ft. long, 9 ft. deep, 14 ft. broad at top, and 11 ft. broad at the bottom; how many cubic feet of water will fill it? If half that quantity of water is supplied, how high will it rise? 5. How many cubic feet of water are contained in a ditch, shaped like the frustum of a wedge, 120 yds. long, 6 ft. deep, 10 yds. broad at the top, and 4 yds. at the bottom. |