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SOME USEFUL RULES,
FINDING THE CONTENTS OF SUPERFICIES AND

SOLIDS.
SECTION I. OF SUPERFICIES.
superficies or area of any plane surface, is com-
pr made up of squares, either greater or less, ac.
3 to the different measures by which the dimen-
f the figure are taken or measured :-and because
nes in length make 1 foot of long measure, there
2x12=144, the square inches in a superficial foot;

Or, in measuring boards, you ma
in feet by the breadth in inckes, :
quotient will give the answer in sqa

Thus, in the foregoing example,
4. If a board be 8 inches wide,
will make a square foot ?
Rule.--Divide 144 by the breadt

-. I. To find the area of a square having equal

5. If a piece of land be 5 rods wide length will make an acre ?

Rule.-Divide 160 by the breadt will be the length required, thus, 5) 1

RULE. tiply the side of the square into itself, and the ct will be the area, or content.

EXAMPLES. How many square feet of boards are contained in or of a room which is 20 feet square ?

20 X 20=400 feet, the Answer. Buppose a square lot of land measures 26 rods on

de, how many acres doth it contain ? TE.-160 square rods make an acre. erefore, 26X26=676 sq. rods, and 676--160=4&.

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

RULE. ltiply the length by the breadth, and the product e the area, or superficial content.

EXAMPLES. A certain garden, in form of a long square, is 96 ft. and 54 wide; how many square feet of ground are ined in it? Ans. 96X545144 square feet. A lot of land, in form of a long square, is 120 rods Agth, and 60 rods wide; how many acres are in it? X60=7200 sqr. rods, then ?200 = 45 acres. Ans. 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 21x1,5=31,5 Ans.

ART. 3. To measure a 'T Definition.- A Triangle is ary thir which is bounded by three right lines.

RULE.
Multiply the base of the given tri
perpendicular height, or half the base
pendicular, and the product will be the

EXAMPLES.
1. Required the area of a triangle w
side is 32 inches, and the perpendicul:

32x7 224 square in
2. There is a triangular or three c
whose base or longest side is 51. rods
from the corner opposite the base mea
many acres doth it contain?

51,5 x 29=1133 square rods,=

*A Triangle may be either right an either case the teacher can easily give idea of the buse and perpendicular, by nslate, paper, ge.

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Or, in measuring boards, you may multiply the length in feet 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 be 8 inches wide, low 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 15 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 wliose 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 obligue ; im either case the teacher can easily give the scholar a right idea of the base and perpendicular, by marking it down on a slate, paper, $€.

TO MEASURE A CIRCLE. ART. 4. The diameter of a Circle being given, to find the Circumference.

. RULE. As 7 : is to 22 : : so is the given diameter : to the circumference. Or, more exactly, As 113 : is to 535 : : &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 diamcter is 4 feet?--As 7 : 22 :: 4 : 12,57 the circumfe. rence.

2. What is the circumference of a circle whose diame. ter is 35 ?-As 7 : 22 : : 35 : 110 Ans.--and inversely as 22 : 7 :: 110 : 35, the diaineter, &c. . Art. 5. To find the area of a Circle.

RULE.. Multiply half the diameter by half the circumference, and the product is the area ; or if the diameter is given without the circumference, multiply the square of the diameter by ,7854 and the product 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 diame ter is 11 rods ?

97854 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 12x12x12 make 1 cubior solid foot.

.

12

Art. 6. To measure a Cabe. Definition.-A cube is a solid of six equal sides, each of which is an exact square.

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

EXAMPLES. 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. fto ... 1 6=1,5 and 1,5X1,5X1,5–3,375 solid feet. Ans.

Or,. 18x18x18=5852 solid inches, and =3,375

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

: 12x12x12=1728 solid 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.
1.1. 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 = 575 foot.

Prod. 1,125x9=10,125 solid feet, the Ans.

in. in. in solid in.
Or, 18x9x108=174967-1728=10,125 feet.

But, in measuring timber, you may multiply the breadth in inches, and the depth in inches, and that product by the length in feet, and divide the last product by 144, which will give the solid content in feet, &c.

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

Breadth 16 inches.
Depth 11

; Prod. 176x20=3520 then, 3520-144=24,4 fecith

the Answer. 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 ?

11x8=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 make a solid foot ? 18x14=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 on timber, of equal bigness from end to end. :

RULE. 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 by the length, and the prodøct will be the solid content. is

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

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