MENSURATION 266. Ratio of circumference to diameter. By measuring several circles, dividing each circumference by its diameter, and taking the average of the results, the circumference will be found to be about 34 times the diameter. 267. Value of π. It is proved in Geometry that this ratio of circumference to diameter is more nearly 3.1416. The ratio is denoted in mathematics by the Greek letter π (pī). 268. Formula for circumference. Therefore, if c = circumference, d = diameter, and r = radius, we have whence с d c = πd, or, because d = 2 r, C = TX 2 r = 2 πr 269. Illustrative problems. 1. Required the circumference when the radius is 7 in. c = 2 πr = 2 × 3 × 7 in. = 44 in. More exactly, 2 × 3.1416 × 7 in. = 43.9824 in. 2. Required the diameter when the circumference is 2827.44 in. Using π = 31 = 22, state the circumferences of circles of diameters as follows: WRITTEN EXERCISE Find the circumference (π = 31), given the diameter: Find the diameter (3), given the circumference: Find the circumference (T=3.1416), given the diameter: Find the diameter (T=3.1416), given the circumference: Find the circumference (π = 31), given the radius: 40. What is the diameter of a water tank whose circum ference is 74.8 ft.? (π = 34.) 41. What is the circumference of a rod whose diameter as measured by the calipers is 2.1 in.? (= 34.) 42. What is the circumference of a wire whose diameter as measured by the calipers is 0.63 in.? (π = 34.) 43. What radius must be used in drawing a pattern for a wheel that shall be 914 in. in circumference? (π = 34.) 270. Area of a circle. A circle can be separated into figures which are nearly triangles. The height is the radius, and the sum of the bases is the circumference. If these were exact triangles the area would be × r × c, or 1 rc. It is proved in Geometry that this is the true area. 271. Given the radius, to find the area. If a = cumference, and r = radius, area, c = cir 'a = } re, or, because e is the same as 2 πr, we may write a = 3 r × 2 πr, or a = rx 2 πr = πr2. 272. The area of a circle is π times the square on the radius. 273. Illustrative problem. Required the area of a circle whose radius is 5 in. a = m2 = = x 25 sq. in. = 3.1416 × 25 sq. in. = 78.54 sq. in. In the rest of the problems involving π, use the value 34 unless otherwise directed. ORAL EXERCISE State the area, given the radius as follows: 1. 7 in. 2. 70 in. 3. in. 4. in. 5. If you take a radius half as long, the area will be what part as great? 6. If you double the length of the radius, the area will be how many times as large? 274. Illustrative problem. Required the radius, the area 1. What is the area of the cross section of a water pipe that is 2 in. in diameter? 2. How many square feet in the base of a water tank that is 42 ft. in diameter ? 3. A horse tethered by a rope 21 ft. long can graze over how many square feet of ground? 4. A horse tethered by a rope can graze over 3850 sq. ft. of ground. How long is the rope? 5. How long is the equator on a globe 12.67 in. in diameter? What is the area of the equator circle cut from such a globe? 6. A school flag pole has a circumference of 24.2 in. at the base. What is the diameter? the radius? the area of a cross section? 7. What is the area of the cross section of a circular iron beam whose circumference is 31.416 in.? (Use π=3.1416.) Suppose the circumference were 53.4072 in.? 8. A boy has a ball tied to a string 1 yd. 6 in. long. As he swings it around, how long is the circumference traveled by the ball? What is the area of the circle inclosed? 9. A tinsmith wishes to make a pattern for the bottom of a pail, the area being 154 sq. in., and to allow in. all around for soldering. What radius should he use in drawing the circle? ORAL EXERCISE 1. A represents a cube 1 in. on an edge. What is its volume? Suppose it were 2 in. on an edge? A 2. B represents half of a cube that was 1 in. on an edge. What is its volume? Suppose the original cube were 3 in. on an edge? 3. What is the volume of a prism 1 in. high, with a base sq. in.? 1 sq. in.? 3 sq. in.? B. 4. Suppose the area of the base in C is 5 sq. in., what is the volume of the lower shaded part that is 1 in. high? What is the total volume? 275. Volume of a prism. We see that, if b is the area of the base of a prism, h is the height, and v is the volume, v = bh. Pupils should be led to read their own rules from all such formulas. Thus, 276. The volume of a prism equals the area of the base multiplied by the height. This is to be understood with the meaning given in § 38. State the volumes of prisms with bases and altitudes as follows: 5. 9 sq. in., 4 in. 7. 16 sq. in., 5 in. 9. 480 sq. in., 121⁄2 in. 11. 330 sq. in., 33 in. 13. 8000 sq. in., 75 in. 6. 17 sq. in., 7 in. 8. 440 sq. in., 25 in. 10. 900 sq. in., 50 in. 12. 480 sq. in., 163 in. 14. 500 sq. ft., 10 ft. |