8. What is the side of a square equal in area to a circular plat 50 feet in diameter? Ans. 44.31 feet. 9. What must be the length of a tether fastened to a horse's neck, that it may sweep over just one acre? Ans. 7.136+ rods. 10. How large a square can be cut out of a circular piece of plank 300 inches in circumference? 11. How many rods in length must be a rope, such as, with one end fastened to a stake in a meadow, and the other to the nose of a cow, will allow her to graze over just 2 acres? PRISMS AND CYLINDERS. 421. A Prism is a volume having two faces equal and parallel polygons, and the other faces parallelograms. The BASES of a prism are its equal and parallel polygons. The CONVEX SURFACE of a prism is formed of its lateral faces, or parallelograms. The prism is triangular, quadrangular, etc., according as its base is a triangle, quadrilateral, etc. Thus, K F G H B E The diagram represents a pentangular prism, whose bases are A B C D E, and FGHIK, and whose convex surface is formed by the faces A B G F, BCH G, etc. A B 422. A Cylinder is a round body of uniform diameter, whose bases are equal and parallel circles. The ALTITUDE of a cylinder, or of a prism, is the straight line joining the centers of the two bases. Thus, The diagram represents a cylinder, of which A B is the altitude. The Bases of a prism? The Convex Surface? A What is a Prism? Cylinder? The Altitude? 423. By Geometry, there may be established the following PRINCIPLES. 1. The CONVEX SURFACE of a prism is equal to the product of the perimeter of the base by the altitude. 2. The CONVEX SURFACE of a cylinder is equal to the product of the circumference of the base by the altitude. 3. The ENTIRE SURFACE of a prism or cylinder is equal to the convex surface plus the area of the bases. 4. The CONTENTS of a prism or cylinder are equal to the product of the area of the base by the altitude. Exercises. 1. What is the entire surface of a square prism whose side is 4 feet wide and length 30 feet? SOLUTION. 30 X 4 X 4 = 480; 4 X 4 X 2 = 32; 480 +32=512 sq. ft., Ans. 2. Required the convex surface of a roller 4 feet in diameter and 10 feet long? Ans. 125.66+ sq. ft. 3. Required the contents of a cylinder 90 centimeters in diameter and 10 meters in length. Ans. 6.36+ cubic meters. 4. If each side of the base of a triangular prism be 2 inches and its length 14 inches, what are its contents? 5. What are the contents of a stick of timber 22 feet 7 inches long, 1 foot 5 inches broad, and 61⁄2 inches thick? Ans. 17.329 cu. ft. PYRAMIDS AND CONES. 424. A Pyramid is a body whose base is any polygon, and whose other faces are triangles meeting at a common point. To what is the convex surface of a prism equal? The convex surface of cylinder? The entire surface? The contents of a prism or cylinder? What is a Pyramid? The VERTEX is the common point at which the triangular faces meet. The CONVEX SURFACE is formed of the triangular faces. The diagram represents a pentangular pyramid, whose vertex is S, and whose convex surface is formed by the faces ASB, BS C, CSD, etc. 425. A Cone is a body whose base B A is a circle, and whose convex surface tapers uniformly to a point at the top, or vertex. The ALTITUDE of a pyramid or cone is a straight line drawn from the vertex perpendicular to the base. Thus, In the diagram the line AB represents the altitude of a cone. The SLANT HIGHT of a pyramid or cone is the shortest straight line that can be drawn from the vertex to the perimeter or circumference of the base. Thus, C In the diagram the line A C represents the slant hight of the cone. 426. The Frustum of a pyramid or cone is the part which remains after cut ting off the top by a plane parallel to the base. Thus, The diagram CDEF represents the frustum of a cone. 427. By Geometry, there may be established the following What is the Vertex of a pyramid? How is the Convex Surface formed? What is a Cone? The Altitude of a pyramid or cone? The Slant Hight? The Frustum of a pyramid or cone? PRINCIPLES. 1. The CONVEX SURFACE of a pyramid or cone is equal to the product of the perimeter or circumference of the base by half the slant hight. 2. The ENTIRE SURFACE of a pyramid or cone is equal to the convex surface plus the area of the base. 3. The CONVEX SURFACE of the frustum of a pyramid or cone is equal to half the product of the sum of the perimeters or circumferences of the two bases by the slant hight. 4. The ENTIRE SURFACE of a frustum of a pyramid or cone is equal to the convex surface plus the areas of the two bases. 5. The CONTENTS of a pyramid or cone are equal to the product of the area of the base by one third of the altitude.. 6. The CONTENTS of a frustum of a pyramid or cone are equal to the sum of the areas of the two bases plus the square root of their product, multiplied by one third of the altitude. Exercises. 1. What is the surface of a square pyramid, each side of whose base is 3 feet, and the slant hight 24.05 feet? Ans. 153.3 sq. ft. 2. Required the number of yards of canvas that will cover a conical tent the slant hight of which is 20 feet and circumference of the base 60 feet. Ans. 66 sq. yd. 3. If the slant hight of a frustum of a triangular pyramid is 12 decimeters, each side of the one base 15 decimeters, and of the other base 9 decimeters, how many square meters is its entire surface? Ans. 5.6449 sq. meters. 4. If one of the largest of the Egyptian pyramids is 477 feet in slant hight, and each side of its base, which is square, is 720 feet, what are the contents in solid yards? Ans. 2003200 cu. yd. To what is the convex surface of a pyramid or cone equal? The entire surface? The convex surface of the frustum of a pyramid or cone? The entire surface? The contents of a pyramid or cone? Of a frustum of a pyramid or cone? 5. Required the number of cubic feet in a conical stack of hay whose hight is 21 feet and the diameter of whose base is 9.5 feet. Ans. 496.176 cu. ft. 6. The diameter of the larger end of a round spar is 30 inches, that of the smaller end 18 inches, and the length 45 feet; required its contents. Ans. 144.31+ cu. ft. 7. If the length of a stick of timber, in the form of the frustum of a pyramid, be 18 feet 8 inches, the side of its larger end 27 inches, and that of its smaller 16 inches, how many cubic feet are there in it? Ans. 61.228 cu. ft. SPHERES. Ꭰ 428. A Sphere is a volume bounded by a curved surface, all points of which are equally distant from a point within called the center. 429. The RADIUS of a sphere is a straight line drawn from the center to any point in the surface. 430. The DIAMETER of a sphere is a straight line drawn through its center, and termi nated both ways by the surface. A Thus, C B E In the diagram the line C B denotes the radius and D E the diameter of a sphere. By Geometry, there may be proved the following PRINCIPLES. 1. The SURFACE of a sphere is equal to the product of the REVIEW QUESTIONS. What is Mensuration? (409) A Triangle? (410) A Quadrilateral? (412) A Circle? (418) A Prism? (421) A Pyramid? 424) A Cone? (425) - What is a Sphere? The Radius of a sphere? The Diameter ? |