MEASUREMENT OF RECTANGULAR SURFACES.

6. What is a Rectangular Figure ?

A Rectangular Figure is one which has four sides and four right angles. (See next Fig.)

When all the sides are equal, it is called a square, when the opposite sides only are equal, it is called an oblong, or parallelogram.

7. What is the area of a figure?

The Area of a Figure is the quantity of surface it contains. It is often called the superficial contents. NOTE.-The term rectangular signifies right angled.

1. How many square feet of canvas in a rectangular painting 3 feet long and 2 feet wide?

feet.

2

ILLUSTRATION.-Let the painting be represented by the figure in the margin; its length being divided into three equal parts, and its breadth into two; each denoting a linear foot. It will be seen that there are 2 rows of squares in the figure, and 3 squares in a row. Therefore, the painting must contain 2 times 3, or 6 square feet.

8. How find the area of a rectangular surface? Multiply the length and breadth together.

3 feet.

2. How many square feet in a blackboard 8 ft. long and 34 ft. wide?

3. How many square inches in a pane of glass 32 in. long and 24 in. wide?

4. How many square rods in a garden 15 rods long and 6 rods wide?

5. How many yards of carpeting 1 yard wide are required to cover a room 18 feet long and 15 feet wide?

6. How many sq. feet in a board 16 ft. long and 1 ft. wide?

7. How many acres in a farm 100 rods long and So rods wide?

8. How many acres in a township 6 miles square?

9.

A flower garden is 30 yards long and 18 yards wide what are its contents?

10. How many brick 8 in. long and 4 in. wide, will i take to pave a sidewalk 40 ft. long and 5 ft. wide?

11. What is the cost of a pine board 18 ft. long and 2 ft. wide, at 8 cts. a square foot?

MEASUREMENT OF RECTANGULAR SOLIDS.

9. What is a Rectangular Body? A Rectangular Body is one bounded by six rectangular sides, each opposite pair being equal and parallel; as, boxes of goods, blocks of hewn stone, etc.

2 feet.

When all the sides are equal, it is called a cube.

10. What are the Contents of a body?

The Contents or Solidity of a body is the quan tity of matter or space it contains.

1. How many cu. feet are there in a box of books 4 ft. long, 3 ft. wide, and 2 ft. deep?

ILLUSTRATION.-Let the box be represented by the preceding figure; its length being divided into four equal parts, its breadth into three, and its depth into two; each part denoting a linear foot. In the upper surface of the box there are 3 times 4, or 12

sq. feet. Now, if the box were I foot deep, it would contain I time as many cubic feet as there are square feet in its upper face, and I time 4 × 3=12 cu. ft. But the box is 2 feet deep; therefors it must contain 2 times 4× 3=24 cu. feet.

11. How find the contents of a rectangular body.

Multiply the length, breadth, and thickness together.

2. How many cu. inches in a brick 8 in. long, 4 in. wide, and 2 in. thick?

3. How many cubic feet in a box of sugar 5 ft. long, 3 ft. wide, and 3 ft. deep?

4. Henry made a pile of cubic letter blocks, the length of which was 8 blocks, the width 6 blocks, and the height 5 blocks: how many blocks were in the pile? 5. How many cubic feet in a pile of brick 13 ft. long, 7 ft. wide, and 5 ft. high?

6. How many cubic feet in a load of wood 7 ft. long, 4 ft. wide, and 3 ft. high?

7. How many cu. feet in a bin 12 ft. long, 6 ft. wide, and 5 ft. deep?

8. How many cu. yards of earth must be removed to dig a cellar 36 ft. long, 20 ft. wide, and 6 ft. deep? 9. How many cu. feet in a stick of timber 36 ft. long, I ft. wide, and 1 ft. thick?

10. What will it cost to build a wall 120 ft. long, 14 ft. thick, and 9 ft. high, at 15 cts. a cubic foot?

11. What will it cost to dig a trench 100 ft. long, 9 ft. wide, and 4 ft. deep, at 27 cts. a cu. yard?

12. What is the worth of a pile of wood 48 ft. long, 6 ft. high, and 4 ft. wide, at $4 a cord?

13. A rectangular bin is 10 ft. long, 6 ft. wide, and 4 feet deep: what are its contents?

14. A load of wood is 7 feet long, 4 ft. wide, and 3 ft. high: what are its contents?

REDUCTION OF DENOMINATE FRACTIONS.

12. What is a Denominate Fraction?

A Denominate Fraction is one or more of the equal parts into which a Compound or Denominate number may be divided.

13. How are they expressed?

Denominate Fractions are expressed either as common fractions, or as decimals; as, 4 pound, .8 yard.

To reduce Denominate Fractions to Units of Lower Denominations.

1. Reduce gallon to quarts and pints.

ANALYSIS. Since there

OPERATION.

are 4 qts. in a gal., there g. ×4=%, or 1 qt. and qt. rem.

must be 4 times as many quarts as gallons; and 4 times gal. are , equal to I qt. and

qt. x2=g, or 1 pt.

Ans. 1 qt. 1 pt.

qt. rem. Again, since there are 2 pt. in a quart, there must be 2 times or pt., equal to 1 pt. Therefore, etc.

14. How reduce denominate fractions to units of a lower denomination?

I. Multiply the given numerator by the number required to reduce the fraction to the next lower denomination, and divide the product by the denominator. (P. 173, Q. 4.)

II. Multiply and divide the successive remainders in the same manner till the lowest denomination is reached. The several quotients will be the answer required.

2. Reduce

of a yard to feet and inches.

3. Reduce of a pound sterling to shillings and pence. 4. Reduce of a week to days and hours.

5. What part of a pint is of a gallon?

SOLUTION. This example is the same in principle as the preceding. Reducing the numerator to the required denomination, place it over the given denominator: gal. 4×2=f, or } pt.

6. What part of a quart is of a bushel?

7. What part of a pennyweight is 30 pound Troy? 8. Reduce .6 yard to feet and inches.

ANALYSIS.-Reducing .6 yard to feet, we have 6 yd. x3=1 ft. and .8 ft. over. Again, reducing .8 ft. to inches, we have .8 ft. x 12=9.6 in. Therefore, .6 yard equals 1 ft. 9.6 in., which is the answer required. Ans. 1 ft. 9.6 in.

15. How reduce a denominate decimal to units denominations?

.6 yd.

3

1.8 ft.

12

9.6 in.

of lower

I. Multiply the denominate decimal by the number required of the next lower denomination to make one of the given denomination, and point off the product as in multiplication of decimals.

II. Proceed in this manner with the decimal part of the successive products, as far as required. The integral part of the several products will be the answer.

NOTE. The preceding operations in Denominate Fractions are the same in principle as those in Reduction Descending.

9. Reduce .84 gal. to quarts and pints.

10. Reduce .625 week to days, etc.

11. Reduce .875 bushel to pecks, etc.

To reduce a Compound Number from a Lower to a Denominate Fraction of a Higher Denomination.

12. What part of a gallon is 1 pint and 2 gills?

ANALYSIS.-I pt. 2 gi. = 6 gills;

and I gallon=1×4×2×4=32 gills. But 6 gills are of 32 gi., equal to gal. Therefore, etc.

1 pt. 2 gi.=6 gi. 1 gal. × 4 × 2 × 4=32 gi.

3

Ans. 32, or 1 gal.

16

16. How reduce a compound number to a denominate fraction of a higher denomination?

I. Reduce the given number to its lowest denomination for the numerator.

II. Reduce to the same denomination, a unit of the required fraction, for the denominator.