Anf. 150 Ex. 2. Required the folidity of a parallelopipedon, whose length is 45 feet, breadth 10 feet, and depth 5: feet. Ans. 2495 cubic feet. Ex. 3. Required the solidity of a parallelopipedon, whose three dimensions are 301, 4., and 2 feet. Anf. 2891 folid feet. Ex. 4. What is the solid content of a parallelopipedon, whose length is 25, breadth 3, and thickness 2 feet? Ex. 5. Required the solidity of a triangular prism, whose length is 10 feet, one side of its triangular base being 14 inches, and the p.rpendicular falling upon it from the opposite angle, 10 inches. Ans. 5 feet, 1 inch 3 parts. Ex. 6. Required the solid content of a pentagonal prism, whose length is 20 feet, and side so feet. Ans. 3440.95 feet. Ex: 7. The same dimensions being given, required the solidity of an octagonal prism. Ans: 9656.854 cubic feet. Ex. 8. On the same supposition, required the solidity of a decagonal prism. Anf. 15388.41 solid feet. Note, From the foregoing examples it is evident, that the nearer the figure of the base approaches to a circle, the greater will the folidity be. Аа Ex. 1 1 Ex. 9. Required the solidity of a cylinder, the diameter of its bafe being 15 inches, and length 14 feet. Anf. 17.180625 cubic feet. Ex. 10. What is the solidity of a pillar 60 inches diameter, and 56 feet high? Anf. 1099.56 cubic feet. PROBLEM V. To find the superficies of any Pyramid or cone. RULE. Multiply the primeter of the bafe by one half of the flant altitude, to the product add the area of the base, the sum will be the superficies. • The reason of this rule is obvious: For if the base of the pyramid le any relilineal figure, each of the sides will be triangles, whose altitude is the same with the flant altitude of the pyramid. It is also plain, that the convex surface of a cone is the sector of a circle, whose radius is the flant altitude, and arch the circumference of the cone's base. EXAMPLE I. Required the superficies of a right cone, whofe diameter of its base'is 10 feet, and Nant altitude 36 feet. Ex. 2. Required the surface of a square pyramid, the side of the base being 30 inches, and llant altitude 6 feet. Anf. 36 4 19. feet. Ex. 3. If the side of the pentagonal base be 10 inches, and the flant altitude 5 feet, required the surface of the pyramid. Anf. u.61141 feet. Ex. 4. What is the fuperficies of a hexagonal pyramid, whose fide is 15 inches, and flant altitude 4 feet? 1:1f. 19.0594875 81: fect. PROBLEM VI. To find the folidity of a cone, or any pyramid. RULE. Multiply the area of the base by, the perpendicular altitude, and the product will be the folidity. Note. Any pyramid is the third part of a prism of the same base and altitude : Allo a cone is equal to one-third the circumscribing cylinder. EXAMPLE I. Required the folidity of a pentagonal pyramid, whose pere pendicular altitude is 60, and Gide 8 feet. 1.720475 tabular area of a pentagon 64 sq. of the side. 20 one third the perp. alta A a 2 Ex, Ex. 2. What is the folidity of a cone, whose flant altitude is g6 inches, and diameter of its base 20 inches ? Anf. 9998.45616 cubic inches. Ex. 3. Required the folidity of a cone, whose perpendicular height is 5 feet, and diameter of its base 16 inches, Anf. 2.3271 cubic feet. Ex. 4. Required the folidity of a triangular pyramid, its height being 141 feet, and the three sides of its base 12, 14, Anf. 284.13716 19 feet. PROBLEM VII. To find the superficies of the frustum of a cone, or any fyramid, RULE. Add together the primeter of both ends, and multiply ono half the sum by the flant altitude, to the product add the area of both ends, and the sum will be the superficies, EXAMPLE I. Required the surface of the frustum of a square pyramid, the fides of the leffer and greater ends being 14, and 24 inches, and lant altitude 2 feet 3 inches. 14* 14*4=56 the perimeter of the lefser end. 24X4=96 the perimeter of the greater end. Ex. 2. Required the surface of the frustum of a cone, the die ameter at the greater end being 10, at the lesser 6 feet, and fant altitude 15. feet. Anf. 496.3726 sq. feet. Ex. 3. What is the surface of the frustum of a pentagonal pyramid, its flant altitude being 140 inches, and the sides of the ends 20, and 30 inches? Anf. 137.0598 /a• feet. PROBLEM VIII. To find the folidity of the fruftum of a cone, or any pyramid. RULE |