4. Find the area of a regular hexagon one of whose sides is 4 ft. and whose apothem is 3.46 ft. REMARK. Those who desire to do so may use formula of Section 28. Find the area of a regular hexagon one of whose sides is 8 ft., and whose apothem is 6.93 ft. 29. Find the area of a regular octagon one of whose sides is 10 ft., and whose apothem is 12.07 ft. 30. Find the area of a regular pentagon one of whose sides is 4 ft., and whose apothem is 2.75 ft. 31. Find the area of a triangle whose sides are 13 ft., 14 ft., and 15 ft. 32. The base of a rectangle is 48 ft. and the altitude is 36 ft. What is the diagonal? 33. A field in the form of a rhomboid contains 9 A. 96 sq. rd., and the altitude is 32 rd. What is the base? 34. The perimeter of a rectangle is 112 ft. and the width is 24 ft. What is the area? 35. A square lot of land that contains 1 acre is how many feet long? Answer correct to the nearest tenth of a foot. 36. To find the area of trapezium ABCD, find the areas of the triangles of which it is composed. Diagonal AC is 34 ft. long and the perpendiculars DE and BF D upon the diagonal are 4 ft. and 13 ft., respectively. 37. How many board feet are there in a board 16 ft. long, 18 in. wide at one end, and 12 in. wide at the other end? 38. The perimeter of a rectangle that contains 10 A. is 200 rd. Its length is 80 rd.; find its width. 39. The length of a rectangle is to its width as 5 is to 4, and it contains 8 A. Find its length and its width in chains. 40. One side of a field in the form of a trapezoid measures 24 ch. 80 li., another side parallel to it measures 18 ch. 40 li., and the distance between these sides is 12 ch. 60 li. How many acres does the field contain? 41. Find the area of the polygon ABCDE in which AG=2 ch., GH=4 ch., HL=2 ch., LD =1 ch., and the perpendiculars BG=2 ch., CH= 3 ch., and EL = 3 ch. B C G H E NOTE. The area of a polygon may be found by dividing it into right triangles and trapezoids, and taking the sum of their areas. The Circle (For definition of terms and important facts see page 124.) 412. It is proved in geometry that the ratio of the circumference of a circle to the diameter is 3.1416, nearly. The Greek letter π (pi) stands for this ratio; then if the circumference is represented by C, the diameter by D, and the radius by R, (1) C = T. (2) 22 NOTE. Sometimes instead of 3.1416 is used to express the ratio of the circumference to the diameter. 413. A circle may be cut and separated, as here shown, into a series of figures that resemble triangles. www If we should call them triangles, each figure might then be regarded as having the curved side for the base and the radius of the circle for the altitude. The sum of the areas of the figures, or the area of the circle, would be the product of half the number of units in the radius and the number of units in the sum of the bases. It is shown in geometry that the area of the circle is equal to this product; therefore: 414. The area of a circle is equal to the product of half the number of units in the radius and the number of units in the circumference. 415. If C represents the number of units in the circumference, R the number of units in the radius, and S the area of the circle, Or, S=TR2 (the area in terms of the radius). Illustrative Examples 1. Find the area of a circle whose radius is 4 ft. S = 3.1416 × 42, or 50.2656. That is, the circle contains 50.2656 sq. ft. 2. Find the area of a circle whose circumference is 31.416 ft. Explanation: 31.416 ft. = the circumference. C=2πR. R = 31.416 ÷ 6.2832, or 5. .. S=3.1416 × 52, or 78.54. That is, the circle contains 78.54 sq. ft. Exercise 99 Find the areas of circles whose radii are as follows: 1. 6 in. 2. 4 yd. 3. 12 rd. 4. 2 ch. 5. 2 ft. 6 in. Find the areas of circles whose diameters are as follows: 6. 10 yd. 7. 5 ch. 8. 6.4 ft. 9. 2.75 rd. 10. 5.9 ch. |