27. Find the area of a triangle, whose three sides are 12, 16, and 20 yards. Ans. 96 sq. yds.

28. Find the area of a triangle, whose three sides are 20, 30, and 40 poles. Ans. 1 A. 3 R. 10 po.+ 29. How many acres in a triangular piece of land, whose three sides are 20, 32, and 44 chains?

Ans. 29 A. 1 R. 12 po.+

The Trapezium.

VII. To find the area of a trapezium.

RULE.-You will observe that the diagonal divides the figure into two triangles: find the area of each by Case V., their sum will make the total area; or, multiply the diagonal by half the sum of the two perpendiculars for the same result.

All cornered figures may be made into triangles, and the sum of their areas will equal the area of the whole figure.

EXAMPLES.

30. Find the area of a trapezium, whose diagonal is 20 feet, and the perpendiculars severally 8 and 10 feet.

20 × 8280

Ans. 180 sq. feet.

31. Find the area of a trapezium, whose diagonal is 30 feet, and the perpendiculars severally 10 feet 6 inches and 12 feet 6 inches. Ans. 345 sq. feet.

How do you find the area of a trapezium? What is said about all cornered figures?

32. How many acres in a piece of land in the form of a trapezium, when the diagonal is 40 poles, and the perpendiculars severally 20 and 30 poles? Ans. 6 acres 1 rood.

The Hexagon.

The Pentagon.

The Octagon.

VIII. To find the area of a regular polygon.

RULE.-Find the area of one of the triangles by Case V., and multiply it by the number of triangles in the figure; or, multiply the perimeter by half the perpendicular, for the required area; or, multiply the perpendicular by half the perimeter.

The perimeter of a polygon is the sum of all its sides. The perpendicular is a line falling from the center perpendicularly to one of its sides.

EXAMPLES.

33. Find the area of a pentagon whose sides severally are 5 feet, and its perpendicular 4 feet.

5 × 42 10.

Ans. 50 sq. ft.

10 X 5 - 50.

34. Find the area of a hexagon whose sides severally are 6

feet, and its perpendicular 6 feet.

35. Find the area of an octagon whose

feet, and the perpendicular 8 feet 4 inches.

Ans. 108 sq. ft. sides severally are 6 Ans. 200 sq. ft.

How do you find the area of a regular polygon? What is the perimeter of a polygon? What is the perpendicular of a polygon?

IX. The diameter of a circle being given, to find its circum· ference.

RULE. The proportion 1 bears to 3.1416, or 34, or the proportion 7 bears to 22, the diameter bears to the circumference, nearly.

EXAMPLES.

36. What is the circumference of a circle whose diameter is 12 inches? Ans. 37.699 inches.+ 37. The diameter of a circle being 16 feet 6 inches, what is the circumference? Ans. 51.8364 feet. 38. The diameter of a circle being 26 yards, what is the circumference? Ans. 83.2524 yds.

39. The diameter of a circular lawn being 52 yards, how far would a boy run, in running around it ? Ans. 163.3632 yds.

X. The circumference of a circle being given, to find the diameter. RULE. The proportion 3.1416, or 34, bears to 1; or the proportion 22 bears to 7, the circumference bears to the diameter, nearly.

EXAMPLES.

40. The circumference of a circle being 37.6992 inches, what is its diameter? (37.6992 3.1416) Ans. 12 inches.

41. The circumference of a circle being 62 feet, what is its diameter ? Ans. 19.735 ft.+

42. If the circumference of the earth be 25000 miles, what is its diameter ? Ans. 7957 miles.+

The diameter of a circle being given, how do you find the circumference? The circumference being given, how do you find the diameter ?

XI. To find the area of a circle.

RULE.-Multiply half the circumference by half the diameter; or, multiply the square of the diameter by .7854. .7854 is the area of a circle whose diameter is 1.

EXAMPLES.

43. What is the area of a circle whose diameter is 12 inches, and its circumference 371⁄2 inches?

12÷2=6.

371÷2=182.

18x6 Ans. 112 sq. in.

44. What is the area of a circle whose diameter is 10 yards, and its circumference 311 yards? Ans. 78.75 yds.

45. What are the contents of a piece of land, in a circular form, the diameter of which is 40 poles, and the circumference 125.664 poles? Ans. 7 A. 3 R. 16 po.+

The Ellipsis.

XII. To find the area of an ellipsis. RULE.-Multiply the longer diameter by the shorter, and the product by .7854 for the area.

EXAMPLES.

46. Find the area of an ellipsis whose longer diameter is 20 inches, and its shorter 12 inches.

47. Find the area of a park in the form of an ellipsis, 250 Ans. 4 A., or 9 po. 22 yds.

yards long and 100 yards wide.

48. Find the area of a wrought board, 6 feet 6 inches long Ans. 17.86785 sq. ft.

and 3 feet 6 inches wide.

How do you find the area of a circle?
How do you find the area of an ellipsis?

SOLIDS.

The Cube.

The Parallelopipedon.

The Prism.

The Cylinder.

XIII. To find the solidity of a cube, a parallelopipedon, a prism, or a cylinder.

RULE. Find the area of one end, and multiply it by the length.

To make solid feet, if either the length, breadth, or depth be given, in inches, divide by 12; if two of these be inches, divide by 144; if all be inches, divide by 1728.

NOTE. To find the surface of a cylinder, multiply the circumference by the length.

EXAMPLES.

49. Find the solid contents of a cube 4 feet wide, 4 feet high, and 4 feet long. (See I.) Ans. 64 solid feet.

50. Find the solid contents of a parallelopipedon, whose end is 2 feet each way, and the length 12 feet.

2 X 2 = 4. 4 X 12 48. Ans. 48 solid feet.

51. Find the solid contents of a prism, the base of whose triangular end is 6 feet, the perpendicular 4 feet, and the length 10 feet. (See V.) Ans. 120 solid feet.

6 X 42=12. 12 X 10 120.

How do you find the solidity of a cube, a parallelopipedon, a prism, or a

der?