20. 11. The area of a triangle is 4 square feet, and tude is 11 inches. What is its base? 104%d its alti 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? Yo 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?/2 15. The area of a trapezoid is 70 square feet and its 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. 17. The diameter of a circle is 4 feet. What is its circumference 12.5 66 97 04 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? What is its area? What is its area? 21. The area of a circle is 100 square feet. its radius?.55 What is ་ What is its circumference?, 22. The area of a circle is 1000 square feet. 28. What is the difference between a circle whose radius is 10 feet, and a triangle whose base is 10 feet and altitude 16 feet? 26.461 irge b 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 length of the arc? 21. 26. If the circumference of a circle is 27 feet, how long is an arc of that circle containing 60°? 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 5.2715 28. If the radius of the sector A B C (fig. 124) is 3 F 124 feet its arc BC800, and the chord B C 3.3 feet, what is the area of the segment? 2.1540 Note. The altitude A D of the triangle A B C is found by the equation" A D±(B2-1 B c2). For A D falls upon the middle of B C (28), and the square of BC is BC2. 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-. 4. 3. The convex surface of a cone is found by multiplying the circumference of the base by half the side of the cone-. -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 surface of the frustum of a cone is found by adding the side to the greater radius and multiplying 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 circle-. 189. The following questions may be solved 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 26.820€ 6. 166.816=2.s is its convex surface? 3. The radius of the base of a cone is 7 inches and its = side 16 inches. What is the convex surface? What is What is the convex surface? 101.596 81.59824 Note. The side of a cone is the bypothenuse 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, &3.280 the less radius 4 feet, and the side 7 feet. What is the ent convex surface? What is the entire surface? 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? 20 6.4/1073 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 F108 two sides. Thus K D (fig. 108) is the hypothenuse of the right triangle K M D, of which K MIC, is the altitude of the frustum, and M D=C D—1 K, is the difference between the radii. 04. 27. The radius of a sphere being 8 feet, what is its surface258. 24x=8 كيمو 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 9511 nearly 34880 miles, and the altitude of one of the frigid zones is nearly 320 miles. How many square miles are 10. The altitude of one of the temperate zones is near- 11. The altitude of the torrid zone is nearly 3200 miles. How many square miles are there in its surface ? 29616000. Mensuration of "Solids. 190. In articles 139, 143, 145, 146, 155, 156, 157, 158, 159, 161, 162, the following propositions were demonstrated. 1. The solidity of a cube is found by taking one of its sides three times as a factor-. 2. The solidity of a prism or of a cylinder is found by multiplying the area of base by its altitude 3. The solidity of a pyramid or of a cone is found by multiplying the area of its base by one third of its altitude 4.The solidity of the frustum of a pyramid or of a cone is found by adding the solidities of three pyramids or cones the same altitude as the frustum, and having for their respec tive bases, the greater base, the less base, and a mean propor tional between the two-. 5. The solidity of a sphere is found by multiplying its surface by one third of the radius-. The solidity of a spherical sector is found by multiplying the surface of the 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 of a sector and a cone; the sector being that whose base is the zone or convex surface of the segment, and the cone that whose base is the base of the segment, and whose altitude is the radius of the sphère minus or plus the altitude of the segment, according as the segment is less or greater than a hemisphere-. 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-. 8. 191. The following' questions may be solved by the application of the rules in the preceding article. 1. If the side of a cube be 9 inches, how many cubic or solid inches does it contain?729 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. J. 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?/2 5. If a prism contain 900 solid feet, and if its altitude be 20 feet, what is the area of its base? 45. 1725 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 square feet, what is its altitude? 8 8. If the greater base of the frustum of a pyramid be 75 square feet, its less base 60 square feet, and its altitude 20 feet, how many solid feet does it contain? 9. If the radius of the base of a cylinder be 10 feet 1546 and its altitude 20, how many solid feet does it contain 4529 679 10. If a cylinder contain 1000 solid feet, and if the radius of its base be 6 feet, what is its altitude? $4 11. If the radius of the base of a cone be 8 feet and its altitude 30 feet, what is its solidity? 201064 13. If the greater radius of the frustum of a Cone be 9 feet, the less radius 6 feet and the altitude 12 feet, how many solid feet does the frustum contain ? 1414. If the radius of a sphere is 8 inches, what is its solidity? 2!44-74 15. How many cubic miles does the earth contain ? 36000000How many cubic feet? J8. 395 3348 16. The diameter of the moon is 2160 miles. What is its volume or solidity? 527676/62556 17. The solidity or volume of the sun is 337102 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 spherical sector 2 feet, what is the solidity of the sector? 50.79. 19. If the solidity of a spherical sector be 3000 solid F 125 20. If the radius H G (fig. 125) of a sphere be 8 feet, 197.920be F 125 Note. The radius of the base of the segment P F is found thus. P F (H F2—H P2). Now H F is the ra dius of the sphere, and H P is the radius of the sphere minus the altitude of the segment, or the altitude of the cone HF P. 21. If the radius H G (fig. 125) of a sphere be 8 feet, |