6. How many square feet in a plank, of a rectangular form, which is 18 feet long and 1 foot 6 inches wide? 7. How many acres in a rectangular garden, whose sides are 326 and 153 feet? Ans. 1 A. 23 P. 64 yd.

8. A rectangular court 68 ft. 3 in. long, by 56 ft. 8 in. broad, is to be paved with stones of a rectangular form, each 2 ft. 3 in. by 10 in.; how many stones will be required? Ans. 2,062 stones.

9. Required the area of the rhomboid A B CD, of which the side A B is 354 feet, and the perpendicular distance, EF, between A B and the opposite side CD, is 192 feet.

354 X 192 67,968 feet, Ans.

=

D E

C

A

F B

10. How many square feet in a flower-plat, in the form of a rhombus, whose side is 12 feet, and the perpendicular distance between two opposite sides of which is 8 feet?

11. How many acres in a rhomboidal field, of which the sides are 1,234 and 762 links, and the perpendicular distance between the longer sides of which is 658 links? Ans. 8 A. 19 P. 4 yd. 61 ft.

PROBLEM II.

614. The area of a SQUARE being given, to find the side. Extract the square root of the area.

Scholium. This and the two following problems are the converse of Prob. I.

EXAMPLES.

1. What is the side of a square containing 625 square feet?

625 25 feet, the side required.

2. The area of a square farm is 124 A. 1 R. 1 P.; how many links in length is its side?

3. A certain corn-field in the form of a square contains

15 A. 2 R. 20 P. If the corn is planted on the margin, 4 hills to a rod in length, how many hills are there on the margin of the field? Ans. 800 hills.

PROBLEM III.

615. The area of a RECTANGLE and either of its sides being given, to find the other side.

Divide the area by the given side, and the quotient will be the other side.

EXAMPLES.

1. The area of a rectangle is 2,072 feet, and the length of one of the sides is 56 feet; what is the length of the other side?

2072 56 37 feet, the side required.

2. How long must a rectangular board be, which is 15 inches in width, to contain 11 square feet?

3. A rectangular piece of land containing 6 acres is 120 rods long; what is its width? Ans. 8 rods. 4. The area of a rectangular farm is 266 A. 3 R. 8P., and the breadth 46 chains; what is the length?

PROBLEM IV.

Ans. 58 chains.

616. The area of a RHOMBOID or RHOMBUS and the length of the base being given, to find the altitude; or the area and the altitude being given, to find the base.

Divide the area by the length of the base, and the quotient will be the altitude; or divide the area by the altitude, and the quotient will be the length of the base.

EXAMPLES.

1. The area of a rhomboid is 67,968 square feet, and the length of the side taken as its base 354 feet; what is the altitude?

67,968354192 feet, the altitude required.

2. The area of a piece of land in the form of a rhombus

is 69,452 square feet, and the perpendicular distance between two of its opposite sides is 194 feet; required the length of one of the equal sides. Ans. 358 ft.

3. On a base 12 feet in length it is required to find the altitude of a rhombus containing 968 square feet.

4. The area of a rhomboidal-shaped park is 1A. 3R. 34P. 5 yd.; and the perpendicular distance between the two shorter sides is 96 yards; required the length of each of these sides? Ans. 18 rods.

PROBLEM V.

- 617. The diagonal of a SQUARE being given, to find the

area.

Divide the square of the diagonal by 2, and the quotient will be the area. (Prop. XI. Cor. 4, Bk. IV.)

EXAMPLES.

1. The diagonal, A C, of the square A B C D, is 30 feet; what is the area?

302 900; 9002450 square feet,

=

2. The diagonal of a square field is 45 chains; how many acres does it contain ?

3. The distance across a public square diagonally is 27 rods; what is the area of the square?

PROBLEM VI.

618. The area of a SQUARE being given, to find the diagonal.

Extract the square root of double the area.

Scholium. This problem is the converse of the last.

EXAMPLES.

1. The area of a square is 450 square feet; what is its diagonal?

450 X 2900; 900 30 feet, the diagonal required.

2. The area of a public square is 4 A. 2 R. 9 P.; what is the distance across it diagonally?

3. The area of a square farm is 57.8 acres; what is the diagonal in chains? Ans. 34 chains.

PROBLEM VII.

619. The sides of a RECTANGLE being given, to cut off a given area by a line parallel to either side.

Divide the given area by the side which is to retain its length or width, and the quotient will be the length or width of the part to be cut off. (Prop. IV. Sch., Bk. IV.)

EXAMPLES.

1. If the sides of a rectangle, ABCD, are 25 and 14 feet, how wide an area, EBCF, to contain 154 square feet, can be cut off by a line parallel to the side AD?

D

A

2. A farmer has a field 16 rods square, and wishes to cut off from one side a rectangular lot containing exactly one acre; what must be the width of the lot?

3. A carpenter sawed off, from the end of a rectangular plank, in a line parallel to its width, 5 square feet. From the remainder he then sawed off, in a line parallel to the length, 8 square feet. Required the dimensions of the part still remaining, provided the original dimensions of the plank were 20 feet by 15 inches.

Ans. 16 feet by 9 inches. 4. The length of a certain rectangular lot is 64 rods, and its width 50 rods; how far from the longer side must a parallel line be drawn to cut off an area of 4 acres, and how far from the shorter side of the remaining portion to cut off 5 acres and 2 roods? How many acres will remain after the two portions are cut off?

PROBLEM VIII.

620. To find the area of a TRIANGLE, the base and altitude being given.

Multiply the base by half the altitude (Prop. VI. Bk. IV.).

621. Scholium. The same result can be obtained by multiplying the altitude by half the base, or by multiplying together the base and altitude and taking half the product.

EXAMPLES.

1. Required the area of the triangle A B C, whose base, B C, is 210, and altitude, AD, is 190 feet.

A

2. A piece of land is in the form of a right-angled triangle, having the sides about the right angle, the one 254 and the other 136 yards; required the area in acres.

Ans. 3 A. 2 R. 10 P. 294 yd.

3. Required the number of square feet in a triangular board whose base is 27 inches and altitude 27 feet.

4. What is the area of a triangle whose base is 15.75 chains, and the altitude 10.22 chains?

5. What is the area of a triangular field whose base is 97 rods, and the perpendicular distance from the base to the opposite angle 40 rods? Ans. 12 A. 20 P.

PROBLEM IX.

622. To find the area of a TRIANGLE, the three sides being given.

From half the sum of the three sides subtract each