11. The area of a triangle is 4 square feet, and its altitude is it inches. 12. What is the difference between'a triangle whose base is 10 feet and altitude 5 feet, and a parallelogram of the same base and altitude? He 13. What is the difference between a triangle whose base is 9 feet and altitude 8 feet, and a square whose side is 6 feet? 14. One of the parallel sides of a trapezoid is 12 inches, the other 16 inches, and the altitude 9 inches. What is the area? 126 15. The area of a trapezoid is 70 square feet and its 20 altitude 7 feet. What is the sum of its parallel sides ? 16. What is the difference between a trapezoid whose altitude is 20 feet and the sum of whose parallel sides is 50 feet, and a triangle whose base is 100 feet and its altitude 10 feet? -0.90c 17. The diameter of a circle is 4 feet. What is its circumference L. 6634 Note. It is sufficient for all common purposes, to multiply by 3.1416 instead of 3.1415926. 18. The circumference of a circle is 75 feet. What is its diameter? What is its radius ? La to. Drug 754.+ 19. The radius of a circle is 7 feet. What is its area? 20. The circumference of a circle is 25 feet. What is its area? 21. The area of a circle is ioo square feet. What is its radius? $55 22. The area of a circle is 1000 square feet. What is its cireumference?. 23. What is the difference between a circle whose radius is 10 feet, and a triangle whose base is 10 feet and altitude 16 feet? 1 6.4.1 Syll 24. The radius of a sector is 4 feet and the arc 12 feet. What is the area ? 24 25. The area of a sector is 90 square feet and the radius 8 feet. What is the lengtla of the arc ? 21 26. If the circumference of a circle is 27 feet, how long is an arc of that circle containing 600 ? 4/4 Note. This is found by the following proportion 360:60::27: answer (15). 27. If the radius of a sector is 5 feet and its arc 700, what is its area - 056275 28. If the radius of the sector A B C (fig. 124) is 3 F 124 feet its arc, B Ç.80, and the chord BC 3.3 feet, what is the area of the segment? 2,154 Note. The altitude A D of the triangle A B C is found Bỳ the equation" # D(**824B c2). For A D falls upon the middle of B C (28), and the square of B C is B C2. 188. In articles 149, 150, 151, 152, 153, the following propositions are demonstrated. 1. «The convex surface of a cylinder is found by multiplying the circumference of its base by its altitude 2. - The entire surface of a cylinder is found by adding the radius of the base to the altitude, and multiplying their sum by the circumference of the base 3. --The convex surface of a cone is found by multiplying the circumference of the base by half the side of the conem 4. -The entire surface of a cone is found by adding the radius of the base to the side of the cone, and multiplying their sum by half the circumference of the base, 5. The convex surface of the frustum of a cone is found by multiplying the side by half the sum of the greater and less circumferences 6. The entire surfdee of the frustum of a corte is found by adding the side to the greater radius and multiplyiny the sum by half the greater circumference; then by adding the side to the less radius and multiplying the sum by half the less circumference; and lastly by adding these two products together 7. -The surface of a sphere is found by multiplying the diameter by the circumference of a great circle. 8. The surface of a zone is found by multiplying its altitude by the circumference of a great circlem, i89. The following questions may be solyed by applying the rules in the preceding article. 1. The radius of the base of a cylinder is 4 inches and its altitude 10 inches. What is its convex surface ? What is its entire surface? 2. The area of the base of a cylinder is 20 square feet and its altitude 8 feet. What is its entire surface? What is its convex surface? 126.120166.81676. 3. The radius of the base of a cone is 7 inches and its 4. side 16 inches. What is the convex surface? What is the entire surface? .551.8: € 408796-am of a cone is 30 square feet and the altitude 10 feet. What is the entire surface ? What is the convex surface ? 106.596.1 111.548 Note. The side of a cone is the hypothenuse of a right triangle, of which the altitude and the radius of the base are the other two sides. 5. The greater radius of the frustum of a cone is 6 feet, 89.280 the less radius 4 feet, and the side 7 feet. What is the enir convex surface ? What is the entire surface ? 11% 91426 6. The greater base of the frustum of a cone contains 40 . square feet, the less base contains 25 square feet, and the altitude is 7 feet. What is the entire surface of the cone? What is the convex surface ? 06.116437 Note. The side of the frustum of a cone is the hypothenuse of a right triangle, of which the altitude and the difference between the greater and less radii, are the other F 108 two sides. Thus K D (fig. 108) is the hypothenuse of the right triangle K MD, of which K M=I C, is the altitude of the frustum, and M D=C D—1 K, is the diffe rence between the radii. 04. ure. The rading of a sphere being 8 feet, what is its sur 8. The diameter of the earth is nearly 7920 miles. Now supposing the figure of the earth to be perfectly spherical, how many square miles are there in its surface? 9. The circumference of a great circle of the earth is gsirm nearly 248807 miles, and the altitude of one of the frigid zones is nearly 320 miles. How many" square miles are there in its surface ? 10. The altitude of one of the temperate zones is nearly 2044 miles. How many square'miles are there in its surface -0755 LVD 11. The altitude of the torrid zone is nearly 3200 miles. How many square miles are there in its surface ? Mensuration of Solids. عو : 190. In articles 139, 143, 145, 146, 155, 156, 157, 5.158, 159, 161, 162, the following propositions were de monstrated. 2. 1. - The solidity of a cube is found by taking one of its c. sides three times as a factor—. - The solidity of a prism or of a cylinder is found by multiplying the area of its base by its altitude 3 3. T'he solidity of a pyramid or of a cone is found by multiplying the area of its base by one third of its altitude 4. T'he solidity of the frustum of a pyramid or of a cone is found by adding the solidities of three pyramids or cones of the same altitude as the frustum, and having for their respective tive bases, the greater base, the less base, and a mean propora tionalsoetween the two 5. — The solidity of a sphere is found by multiplying its surface by One third of the radius 6. The solidity of a spherical sector is found by multiplying the surface of ihe zone, which forms its base, by one third of the radius 7. - The solidity of a spherical segment of one base is found by taking the difference or sum of the solidities af a sector and a cone; the sector being that whose base is the zone or convex surface of the segment, and theicone that whose base is the base of the segment, and whose altitude is the radius of šphère minus or plusitihe altttude of the segment, according as the segment is less or greater than a hemisphere 8. --The solidity of a spherical segment of two bases is found by taking the difference between the solidities of two spherical segments of one base; the respective bases of the latter, being the two bases of the required segment— 191. The folowing questions may be solved by the application of thy rules in the preceding article. 1. If the side of a cube be 9 inches, how many cubic or solid inches does it contain: 72% 2. How many cubic inches are there in a cubic foot ? 3. If a cube contain 2550, solid feet, what is the length of its side. 2.65 Note. This question supposes a knowledge of the process for extracting the cube root of numbers, the explanation of which is generally considered as belonging to arithmetic and algebra. 4. A cord of wood is in the form of a quadrangular prism, 8 feet long, 4 feet wide, and 4 feet high. How many solid feet does it contain 12% 5. If a prism contain 990 solid feet, and if its altitude be 20 feet, what is the area of its base? 457 6. What is the solidity of a pyramid whose base covers a thousand square feet, and whose altitude is 70 feet? 7. If a pyramid contain 800 solid feet and its base 50 is its altitude ? 18 8. If the end greater base of the frustum of a pyramid be 75 square feet, its less base 60 square feet, and its altifude 20 feet, how many solid feet does it contain ? 9. If the radius of the base of a cylinder be 10 feet and its altitude 20, how many solid feet does it contains 10. If a cylinder contain 1000 solid feet, and if the radius of its base be 6 feet, what is its altitude ? 884 11. If the radius of the base of a cone be 8 feet and its altitude 30 feet, what is its solidity ? 12. If the solidity of a cone be 2000 feet and the radi 204084 us of its base 10 feet, what is its altitude de mone be 9 13. If the greater radius of the frustum of a 1117. feet, the less radius 6 feet and the altitude 12 feet, how 678 many solid feet does the frustum contain ? 2741 14. If the radius of a sphere is 8 inches, what is its solidity ? !!44:14 15. How many cubic miles does the earth contain ? 1360608 Alow many cubic feet i J8.381575777461 its volume or solidity ? 11762 8162576 17. The solidity or volume of the sun is 3371,02 times as great as that of the earth. What is the surface of the sun, supposing it spherical? What is its diameter ? 18. If the radius of a sphere be 6 feet, and the altitude of a zone forming the base of a spherica sector 2 feet, what is the solidity of the sector? 150.7 *19. If the solidity of a spherical sector be 3000 solid 'feet, and its radius 50 feet, what is the surface of its zone or base? 160.1% F125 20. If the radius H G (fig. 125) of a sphere be 8 feet, and the altitude P G of the segment P F G of one base, be 3 feet, what is the solidity of the segment PFG? Note. The radius of the base of the segment P F is found thus. PF=(u f2-4 p2)é. Now H F is the radius of the sphere, and H P is the radius of the sphere minus the altitude of the segment, or the altitude of the cone HFP. F 125 21. If the radius H G (fig. 125) of a sphere be 8 feet, 197.920be |