10. What is the area of a trapezium, one of whose diagonals is 34 inches, and the lengths of the two perpendiculars let fall upon it, from the opposite vertices of the trapezium, are, respectively, 16 and 18 inches?

11. Find the area of a trapezium whose sides are successively 35.8, 13.32, 35.84, and 17.8 meters, and the diagonal drawn from the beginning of the first to the end of the second side 38.9 meters.

474. The area of any polygon can be found by dividing the polygon into triangles, and finding the sum of their areas.

NOTE 1. In finding the area of a polygon by this method the lengths of the sides and of the diagonals are supposed to be known.

12. The sides of a field taken in order are 20, 30, 25, 18, and 27 rods, and the two diagonals from the beginning of the first to the extremities of the third side are, the one to the beginning of the third side, 40 rods, and the one to the end of the third side, 35 rods. Find the area of the field.

NOTE 2. In all cases of this kind the figure ought to be drawn.

BOARD MEASURE.

475. A Board Foot is a square foot one inch in thickness. 476. Lumber, that is, sawed timber, boards, planks, joists, and the like, is measured in board feet.

13. How many feet, board measure, in a plank 14 feet long, 16 inches wide, and 3 inches thick?

14. Find the number of feet, board measure, in 25 boards, each 12 feet long, 9 inches wide, and 1 thick.

15. Find the number of feet, board measure, in 55 joists 3 by 4 (that is, the two smaller dimensions of the joists are 3 inches and 4 inches), each 16 feet long.

477. Boards less than an inch in thickness are reckoned the same as though they were an inch in thickness.

16. How many feet, board measure, in 75 boards, each 18 feet long, 8 inches wide, and inch thick? Ans. 900 bd. ft.

17. How many feet, board measure, in 45 boards, each 15 feet long, 10 inches wide, and inch thick?

478. A right-angled triangle is one that B

has a right angle; as ABC.

The side BC opposite the right angle is called the hypothenuse; the other two sides, the base and perpendicular.

C

479. The square described on the hypothenuse of a rightangled triangle is equal to the sum of the squares described on the other two sides.

Hence, the square of

either of the two sides which form the right angle is equal to the square of the hypothenuse diminished by the square of the other side. This will be seen by counting the small squares in the square on the hypothenuse and those in the squares on the other two sides. Hence,

480. To find the hypothenuse when the base and perpendicular are given, add the square of the base to the square of the perpendicular, and find the square root of the sum.

481. To find either side about the right angle when the hypothenuse and the other side are given, from the square of the hypothenuse subtract the square of the other given side, and find the square root of the remainder.

18. The base of a right-angled triangle is 9 feet and the perpendicular is 12 feet; what is the hypothenuse?

√92 + 122 = √225 = 15.

Ans. 15 ft.

19. The hypothenuse of a right-angled triangle is 25 and the base is 15; what is the perpendicular?

20. What is the distance on the floor from one corner to the opposite corner of a rectangular room 16 by 24 feet?

21. A wall 26 feet high has in front of it a ditch 14 feet wide. How long a ladder will it take to reach from the opposite side of the ditch to the top of the wall?

22. What is the length of the longest rod that, without bending, can be put into a box whose inner dimensions are 36, 24, and 12 inches?

23. If a pole 75 feet high is broken off 18 feet from the ground, how far from the foot of the pole will its top strike the ground?

24. How far from a tower 40 feet high must the foot of a ladder 50 feet long be placed that it may exactly reach the top of the tower?

25. If a rope 108 feet long, from a point in the street, will reach on one side to a window 75 feet high, and on the other to a window 45 feet high, how wide is the street?

26. The foot of a ladder 67 feet long stands 40 feet from a wall; how much nearer the wall must the foot be placed that the ladder may reach 10 feet higher?

CIRCLES.

482. The circumference of a circle (Art. 257) is 3.1416 times its diameter; or, the diameter is equal to the circumference divided by 3.1416.

Circumference = 3.1416 x Diameter.

Diameter = Circumference 3.1416.

27. What is the circumference of a circle whose radius is 15 meters?

28. What is the diameter of a circle whose circumference is 57 rods?

29. What is the diameter of a tree whose circumference is 19 feet?

30. What is the circumference of a circular pond whose diameter is 56 meters?

483. The area of a circle is equal to half the product of its circumference and its radius. Or, it is equal to 0.7854 times the square of its diameter.

Area Circumference × Radius.

=

Area = 0.7854 × Diameter 2.

31. What is the area of a circle whose radius is 6 meters?

32. What is the area of a circle whose radius is 40 feet?

33. What is the area of a circle whose circumference is 18 inches?

34. What is the circumference of a circle whose area is 116 square feet?

35. The radii of two concentric circles are 40 and 54 feet; what is the area of the space bounded by their circumferences?

36. A has a circular lot of land whose diameter is 95 rods, and B a similar lot whose area is 750 square rods; compare these lots.

37. What is the difference between the perimeters of two lots of land each containing an acre, if one is a square and the other a circle?

SIMILAR SURFACES.

484. Similar Figures are figures that are of precisely the same form, without regard to their magnitude.

The areas of similar figures are to each other as the squares of their corresponding lines; and, conversely, the corresponding lines of similar surfaces are to each other as the square roots of their areas.

38. A has an acre of land one of whose sides is 20 rods in length; B has a piece of land of exactly similar form containing 169 acres. What is the length of the corresponding side of B's?

OPERATION.

1: 169202: Ans.2, or, 1: 13 = 20: 260. Ans. 260 rods.

39. Of two similar fields one contains 36 acres and the other 72. If a side of the first is 23 rods, what is the length of the corresponding side of the second?

40. The areas of two circles are to each the circumference of the smaller is 18 feet. ference of the greater?

other as 6 to 10, and What is the circum

41. The areas of two similar right triangles are respectively 361 and 225 square feet, and the hypothenuse of the first is 42 What is the hypothenuse of the second triangle?

feet.