EXAMPLES.

65. What is the fourth root of 1048576?

66. What is the sixth root of 17596287801 ?

67. What is the eighth root of 457163239653376 ? 68. What is the ninth root of 520411082988487293 ? 69. What is the twelfth root of 13841287201 ?

70. Find the sixth root of 177978515625.

MENSURATION.

THE TRIANGLE.

[a] (Base X altitude) ÷ 2 = area.

EXAMPLES.

1. What is the area of a triangle whose base is 18 yards and altitude 24 feet?

2. What is the area of a triangle whose base is 90 feet and altitude 40 feet?

3. The base of a triangle is 16 yards and its altitude 8 yards 2 feet. What is its area?

4. How many acres are contained in a triangular piece of land whose base is 120 rods and altitude 132 yards?

5. What is the cost of a triangular piece of ground whose base is 110 yards and altitude 82 yards 1 foot 6 inches, at $64 per acre?

6. What is the area of the gable-end of a house which is 30 feet wide, if the ridge of the roof is 20 feet higher than the foot of the rafters ? base= altitude.

[b] (Area × 2) + { altitude = base.

7. What is the base of a triangle whose area is 16 sq. yd. 8 sq. ft. and altitude 5 yd. 1 ft.?

8. What is the altitude of a triangle whose area is 108 sq. in. and base 1 ft. 6 in. ?

9. What is the base of a triangle whose area is 490 sq. yd. and

altitude 28 yd.?

10. What is the altitude of a triangular piece of ground whose area is 3 acres and base 220 yards?

11. I paid $637.50 for a triangular piece of ground, bought at $75 per If its base is 340 yards, what is its altitude?

acre.

12. What is the area of a triangle whose sides are 18, 20, and 24 feet?

Solution.

(18+20+24) ÷ 2 = 31; 31-1813; 31-2011; 31-247. Square root of (31 x 13 x 11 x 7) = 176.155 + sq. ft., area.

13. The sides of a triangle are 20, 30, and 40 feet. What is its area? 14. How many acres in a triangular piece of ground whose sides are 80, 90, and 120 rods?

15. How many square feet in both gable ends of a house 30 feet wide, if the rafters are 20 feet long?

16. At 50 cents a square yard, what is the cost of paving a triangular court, the length of each side of which is 40 feet?

[c] Base+perpendicular2 = hypothenuse.

17. What is the hypothenuse of a right-angled triangle whose base is 40 feet, and perpendicular 30 feet?

18. The base of a right-angled triangle is 135 yards and its perpendicular 72 yards. What is its hypothenuse?

19. What must be the length of a ladder whose foot is placed 20 feet from a building that it may reach a window 48 feet high?

20. A rectangular field is 84 rods long and 80 rods wide. What is the distance between its opposite corners ?

21. Two pedestrians start from the same point, one going due south at the rate of 3 miles an hour, and the other due east at the rate of 4 miles an hour. How many miles are they apart in 9 hours?

22. What is the distance between one of the lower corners and the opposite upper corner of a room 32 feet long, 24 feet wide, and 30 feet high?

[d] Hypothenuse-base2 = perpendicular.
√

[e] Hypothenuse - perpendicular2 = base.

23. What is the perpendicular of a right-angled triangle, if its base is 45 inches and its hypothenuse 51 inches?

1

24. The base of a right-angled triangle is 105 feet, and its hypothenuse 145 feet. What is its perpendicular?

25. What is the base of a right-angled triangle, if its hypothenuse is 1275 yards and its perpendicular 600 yards?

26. At what distance from a house must the foot of a ladder 78 feet long be placed, that it may reach to a window 72 feet high?

27. The top of a tree was broken off by a storm and struck the ground at a distance of 32 feet from the stump. If the part broken off was 40 feet long, what was the height of the tree?

THE QUADRILATERAL.

[f] Base X altitude = area of any parallelogram.

EXAMPLES.

28. What is the area of a parallelogram whose base is 42 feet and altitude 18 feet?

29. What is the cost of a rectangular piece of land 242 yards long and 110 yards wide, at $36.50 per acre?

30. What is the area of a lot which is 146 yards square?

31. The base of a rhomboid is 18 feet and its altitude 15 feet. What is its area?

32. What is the area of a rhombus whose perimeter is 48 yards and altitude 8 yards?

[g] The area of a trapezoid equals one-half the sum of its two parallel sides X its altitude.

33. Find the area of a trapezoid, the longer of whose parallel sides is 175 feet, the shorter 135 feet, and whose altitude is 80 feet.

34. How many square feet in a tapering plank whose length is 18 feet, and whose width at one end is 8 inches and at the other 6 inches ? 35. The parallel sides of a field are 70 and 90 rods, and its breadth 209 yards. What is the value of the field at $75 per acre?

[h] To obtain the area of a trapezium: Separate it into two triangles, find the area of each triangle, and add the two results.

36. What is the area of a trapezium whose diagonal is 80 feet, and the

altitudes of the two triangles formed by the diagonal are respectively 75 and 45 feet?

37. What is the area of a trapezium whose diagonal is 74 yards, and the altitudes of the two triangles formed by the diagonal are respectively 40 yards and 18 yards?

38. At $48 per acre, what is the cost of a quadrilateral field which measures 130 rods from one of its corners to the opposite corner, and the perpendiculars on this diagonal from the remaining corners measure respectively 26 and 20 rods?

THE CIRCLE.

[i] Diameter x 3.1416 = circumference.

EXAMPLES.

39. What is the circumference of a circle whose diameter is 15 inches? Whose radius is 8 feet?

40. What is the circumference of a circle whose diameter is 8 in. ? Whose radius is 7 yd.?

41. The diameter of a circular pond is 142 feet. What is its circumference?

42. What is the diameter of a circle whose circumference is 8 ft. 3 in. ? Whose area is 1963.5 sq. yd.?

43. The circumference of a circular flower-bed is 13 feet. What is its diameter ?

44. The area of a circular piece of land is 962.115 sq. rd. What is its diameter ?

45. If a wagon-wheel is 6 feet in circumference, what is its diameter ?

46. What is the area of a circle whose diameter is 18 rd.? Whose radius is 25 yd.? Whose circumference is 20 ft.? Whose circumference is 43.9825 in. and diameter 14 in. ?

47. The distance around a circular court is 375 feet. What is its area?

48. If a stake be placed in the centre of a pasture, and a horse be fastened thereto by a halter 15 feet long, over what area will he be able to graze?

49. In the centre of a circular park 300 yards in circumference is a circular pond 145 feet in diameter. How much of the park is land?

50. A circle is 40 feet in diameter, and within it is another circle 15 feet in diameter. What area of the larger circle is outside the smaller?

[1] To find the sides of a square inscribed in a given circle: Multiply the diameter of the circumscribing circle by .7071, or its circumference by .2251.

51. What is each side of the square which can be inscribed in a circle whose circumference is 75 feet? Whose diameter is 18 inches?

52. What is the side of the largest square log which can be hewn from a round piece of timber the small end of which is 2 feet 8 inches in diameter? The small end of which is 4 feet in circumference ?

THE PRISM AND CYLINDER.

[m] { }

Perimeter of the base of a prism

Circumference of the base of a cylinder

Xaltitude convex surface.

53. The altitude of a triangular prism is 8 ft., and the sides of its base are 3 ft., 4 ft., and 5 ft. What is its convex surface?

54. The altitude of a cylinder is 3 ft. 6 in., and the circumference of its base 1 ft. 8 in. What is its entire surface?

55. What is the convex surface of a prism whose altitude is 9 ft., and its ends pentagons each side of which is 5 ft. ?

56. What is the entire surface of a triangular prism whose altitude is 6 ft., and the sides of its ends 3, 4, and 5 ft., respectively?

57. What is the entire surface of a quadrangular prism whose altitude is 18 in., and its ends 12 in. by 15 in. ?

58. What is the entire surface of a cylindrical boiler 2 ft. 9 in. în diameter, and 10 ft. long?

[n] Area of base × altitude = volume of prism or cylinder.

59. What is the volume of a triangular prism whose altitude is 12 ft., and each side of its ends 6 ft. ?

60. What is the volume of a cylinder whose altitude is 15 ft., and the diameter of its ends 4 ft. ?