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SOME USEFUL RULES,

FOR FINDING THE CONTENTS OF SUPERFICES AND SOLIDS.

SECTION 1.-OF SUPERFICES. The superfices or area of any plane surface, is compos sed or made up of squares, either greater or less, ac

according to the different measures by which the dimensions of the figare are taken or measured:-and because 12 ches length make 1 foot of long measure, therefore, 12x12=144 the square inches in a superficial foot, &c.

Art. I. To find the area of a square having equal sides:

RULE. Multiply the side of the square into itself and the product will be the area, or content.

EXAMPLES.

1. How many square feet of boards are contained in the floor of a room which is 20 feet square ?

20 X 20–400 feet, the Answer. 2. Suppose a square lot of land measures 26 rods on each side, how many acres doth it contain ?

NOTE.-160 square rods make an acre.
Therefore, 26 x 26–676 sq. rods, and 676-160=4 a.

36 r. the Answer. Art. 2. To measure a parallelogram, or long square.

RULE. Multiply the length by the breadth, and the product will be the area, or superficial content.

EXAMPLES.

1. A certain garden, in form of a long square, is 96 feet long, and 54 wide; how many square feet of ground are contained in it ? Ans. 96 X 5435184 square feet.

2. A lot of land, in form of a long square, is 120 rods in length, and 60 rods wide ; how many acres are in it?

120 x 60=7200 sq. rods, then 12=45 acres. Ans. 3. If a board or plank be 21 feet long, and 18 inches

how many square feet are contained in it ?

18 inches=1,5 feet, then, 21 x1,5=31,5. Ans.

broad;

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Or, in measuring boards, you may maltiply the length in (set by the breadth in inches, and divide by 12, the quotient will give the answer in square feet, &c.

Thus, in the foregoing example, 21x18:12=31,5 as before.

4. If a board bę 8 inches wide, how much in length will make a square foot ? RULE.--Divide 144 by the breadth, thus, 8)144

Ans. 18 in. 5. If a piece of land be 5 rods wide, how many rods in length will make an acre?

RULE.-Divide 160 by the breadth, and the quotient will be the length required, thus,

5)160

Ans. 32 rods in length. ART. 3.—To measure a triangle. Definition.-A triangle is any three cornered figure which is bounded by three right lines.*

RULE. Multiply the base of the given triangle into half its perpendicular height, or half the base into the whole perpendicular, and the product will be the area.

EXAMPLES.

1. Required the area of a triangle whose base or longest side is 32 inches, and the perpendicular height 14 inches.

32x7=224 square inches the Answer. 2. There is a triangular or three cornered lot of land whose base or longest side is 51} rods; the perpendicular from the corner opposite the base measures 44 rods ; how many acres doth it contain ?

51,5 x 22=1133 square rods,=7 acres, 13 rods.

* A Triangle may be either right angled or oblique; in either case the teacher can easily give the scholar a right idea of the base and perpendicuz las, by marking it down on the slate, paper, &c.

TO MEASURE A CIRCLE,

Art. 4. The diameter of a circle being given, to fini the circumference.

Rule.—As 7 : is to 22 :: so is the given diameter : to the circum ference. Or, more exactly, as 113 : is to 355 : : &c. the diameter is found inversely.

Note.—The diameter is a right line drawn across the circle through its centre.

EXAMPLES.

1. What is the circumference of a wheel whose diameter is 4 feet ?-as 7 : 22 : : 4 : 12,57 the circumference.

2. What is the circumference of a circle whose diameter is 35 ?-As 7 : 22 : : 35 : 110 Ans.—and inversely as

7 : : 110 : 35, the diameter, &c.

22 :

ART. 5.-To find the area of a Circle.

Rule.-Multiply half the diameter by half the circumference, and e product is the area ; or if the diameter is given without the cir. umference, multiply the square of the diameter by ,7854, and the roduct will be the area.

EXAMPLES. 1. Required the area of a circle whose diameter is 12 inches, and circumference 37,7 inches.

18,85–half the circumference.

6=half the diameter.

113,10 area in square

inches. 2. Required the area of a circular garden whose diameter is 11 rods?

,7854 By the second method, 11x11 121

Ans. 95,0334 rods SECTION 2.-OF SOLIDS. Solids are estimated by the solid inch, solid foot, &c. 1728 of these inches, that is, 12 x 12 x 12 make 1 cubic or solid foot:

ART. 6.To measure a Cube. Definition.—A cube is a solid of six equal sides, each of f which is an exact square.

RULE.-Multiply the side by itself, and that product by the same Bide, and this last product will be the solid content of the cube.

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EXAMPLES.

2

1. The side of a cubic block being 18 inches, or 1 foot and 6 inches, how many solid inches doth it contain ? ft. in ft.

1 6=1,5 and 1,5 x 1,5 1,5=3,375 solid feet. Ans. Or, 18 x 18 x 18=5832 solid inches, and 1921–3,375.

2. Suppose a cellar to be dug that shall contain 12 feet of every way, in length, breadth and depth ; how many solid feet of earth must be taken out to complete the same?

12 < 12 x 12=1728 sold feet, the Ans. ART. 7.-To find the content of any regular solid of three

dimensions, length, breadth and thickness, as a piece of timber squared, whose length is more than the breadth and depth.

Rule.-Multiply the breadth by the depth, or thickness, and that product by the length, which gives the solid content.

EXAMPLES.

1. A square piece of timber, being one foot 6 inches, or 18 inches broad, 9 inches thick, and 9 feet or 108 inches long; how many solid feet doth it contain ?

1 ft. 6 in.=1,5 foot 9 inches ,75 foot.

Prod. 1,125 y 9–10,125 solid feet, the Ans.

in. in. solia in
Or 18 x 9 x 108=17496 -- 1728=10,125 feet.

in.

But, in measuring timber, you may multiply the breadth in inches. arzat the depth in ipches, and tha product by che lengto in fee and divide tne ust product by 144, which will give the solid content in fcet, &c.

2. A piece of timber being 16 inchres broad, 11 inches thick, and 20 feet long, to find the content ?

Breadth 16 inches.
Depth 11

Prod. 176 x 20=3520 then, 3520-144=24,4 feet. Ans.

3. A piece of timber 15 inches broad, 8 inches thick, and 25 feet long; how many solid feet doth it contain?

Ans. 20,8 + feet. Art. 8.-When the breadth and thickness of a piece of

timber are given in inches, to find how much in length will make a solid foot. Rule.-Divide 1728 by the product of the breadth and depth, and the quotient will be the length making a solid foot.

EXAMPLES.

1. If a piece of timber be 11 inches broad and 8 inches deep, how many inches in length will make a solid foot ?

11 x8=88)1728(19,6 inches. Ans. 2. If a piece of timber be 18 inches broad and 14 inches deep, how many inches in length will enake a solid foot? 18 x 14=252 divisor, then, 252)1728(6,8 inches. Ans.

ART. 9.--To measure a Cylinder. Definition.--A Cylinder is a round body whose bases are circles, like a round column or stick of timber, of equal big. ness from end to end.

RCLE.-Multiply the square of the diameter of the end by ,7854 which gives the area of the base ; then multiply the area of the base jy the length, and the product will be the solid content,

EXAMPLE

What is the solid content of a round stick of timber of qual bigness from end to end, whose diameter is 18 inches, end length 20 feet?

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