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443. When the three sides of a triangle are known, the area is found as follows:

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II.

III.

Subtract each side separately from half the sum of the three sides.

Find the continued product of the half-sum and the three remainders, and extract the square root of that product. Ex. Find the area of a triangle whose sides are 3, 4, and 5 ft. respectively.

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46. Find the area of a triangle whose sides are 73, 57, and

48 ft.

47. Find the number of hektars in a triangular field whose sides are 37.5m, 91.7m, and 78.9m.

48. Find the number of hektars in a triangular field whose sides are 67.5, 81.2m, and 102.7m.

49. Find the number of acres in a triangular field whose sides are 227, 342, and 416 ft.

50. Find the number of acres in a triangular field whose sides are 79 chains 8 links, 57 chains 3 links, and 102 chains 19 links.

51. Find the number of square rods in a triangle whose sides are 7 rds. 2 yds., 6 rds. 5 yds., and 9 rds. 43 ft. 52. Find the number of acres in a four-sided field, the sides of which are in order 361, 561, 443, and 357 ft.; and the distance from the beginning of the first side to the end of the second side is 682 ft.

444. When one side of a triangle is regarded as a base, the triangle may be imagined as resting with that base on a horizontal line. The distance

of the highest point of the triangle above that line is called the altitude of the triangle.

When the altitude and base of a triangle are known, the area is found as follows:

Multiply the altitude by the base, and take one-half the product.

Ex. Find the area of a triangle of which the base is 3 ft. and altitude 4 ft.

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53. Find the number of hektars in a field of three sides, one of which is 82.1m, and the distance from this side to the opposite corner 47.3m.

54. Find the number of acres in a triangular lot, one side of which is 343.6 ft., and the distance from this side to the opposite corner is 163.2 ft.

When the three sides of a triangle are known, and the altitude is required :

Find the area of the triangle, and divide the result by half the side that is taken as the base.

55. Find the altitude of a triangle, if each side is 1000 ft. 56. Find the distances of the vertices from the opposite sides of a triangle, when these sides are 17.8mm, 23.6mm, and 31.5mm

57.

If the four sides of a field measured in succession are 237, 253, 244, and 261 ft., and the diagonal measured from the end of the first side to the end of the third side is 351 ft.; find its area.

58. If the four sides of a field are 237, 253, 244, and 261 ft., taken in order, and if the corner formed by the second and third sides is a square corner; find the diagonal from the beginning of the second side to the end of the third side, and also find the area of the field. 59. Find the area of a circle that has a radius of 10 in.; of a circle that has a diameter of 10 ft.; of a circle that has a circumference of 30 in.

60. A horse is tied by a rope 27.8m long; what part of a hektar can he graze e?

61. How many square feet in a circle that has a diameter of 17 yds.?

62. How many square feet in a circle that has a circumference of 117 yds.?

63. How many square inches in the surface of a globe that has a radius of 12.37 in.?

64. Find the area of the surface of the largest globe that can be turned out from a joist 4 in. by 6 in. 65. How many cubic inches in a globe that has a diameter of 10 in. ?

66. If a tree be round, and the girt is 17 ft. 6 in., find its diameter. Find the area of a cross-section, and find the number of cubic feet in the largest sphere that can be cut from it.

67. Find the weight in kilograms and in pounds of an iron ball 21.5cm in diameter, specific gravity 7.47; of a tin ball 13cm in diameter, specific gravity 7.29; of a lead ball 17.3cm in diameter, specific gravity 11.35; of a silver ball 1.31cm in diameter, specific gravity 10.47.

68. A slab of cast-iron 4 ft. 2 in. long, 17 in. wide, and 8 in. thick, specific gravity 7.31, is cast into 2-lb. balls. If there is a loss of 5% in melting, how many balls are obtained, and what is the diameter of each?

69. How many pounds avoirdupois would a ball of such iron 30 in. in diameter weigh?

70. If the specific gravity of ice is .921, find the weight and

the surface of each of three spheres of ice whose diameters are 1cm, 10cm, and 1m. Which of these spheres would roll first on a plain, in a graduallyincreasing wind?

445. A straight, round stick, cut off square at each end, is called a cylinder.

The area of the convex surface of a cylinder is obtained as follows: Multiply the circumference of one end by the length of the cylinder.

The volume of a cylinder is obtained as follows:

Multiply the area of one end by

the length of the cylinder..

Cylinder.

71. Given a cylinder 10 in. in diameter and 12 in. long; required the area of each end, the convex area, the total area, and the contents in gallons.

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72. Find the capacity in gallons of a round cistern 13 ft. in diameter and 9 ft. deep.

73. What must be the diameter of a cylinder 10 in. deep, in order that it may hold 1 gallon?

74. Find the volume of a cylinder 8 in. in diameter and 11 in. high.

75. Find the dimensions of three cylinders that have the diameters equal to the heights, and hold 1 gal., 1 qt., and 1' respectively.

446. A solid with two equal polygonal ends, connected by plane faces at right angles to the ends, is called a prism. The volume of a prism is found as follows:

Multiply the area of one end by the length of the prism.

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Prism.

Pyramid.

Cone.

76. Find the volume of a triangular prism 11 in. long, the sides of the ends being 2, 3, and 4 in. long.

77. Find the capacity in bushels of a bin 6 ft. long, and the end of which is a square measuring 3 ft. 3 in. on a side. 78. Find the number of cubic yards in a square prism 200 ft. on a side, and 40 ft. long.

447. A solid with a polygonal base, and plane faces meeting in a point, is a pyramid. The volume of a pyramid is one-third of that of a prism of the same base and height. 79. How many cubic yards in a square pyramid 210 ft. on a side, and 123 ft. high?

80. Find the capacity of a cup, the mouth of which is a square 4 in. on a side, and the sides of which are four equilateral triangles.

81. The largest of the Egyptian pyramids is 147m high, with a base 231m square. Find its volume in cubic

meters.

448. A body whose base is a circle, and whose convex surface tapers uniformly to a point, is called a cone.

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