9* SOME USEFUL RULES, FOR FINDING THE CONTENTS OF SUPERFICIES AND SOLIDS. SECTION I. OF SUPERFICIES. The superficies or area of any plane surface, is coinposed or inade up of squares, either greater or less, according to the different measures by which the dimensiuns of the figure are taken or measured :-and because 12 inches in length make 1 foot of long measure, there fore, 12x19=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 door of a room which is 20 feet square ? 20x20=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. ; 36r. the Answer. ART. 2. To measure a parallelogram, or long square, : RULE. Multiply the length by the breadth, and the prosluct will be the area, or superficial content. EXAMPLES. . 1. A certain garden, in forın of a long square, is 96 ft. ong, and 54 wide; how many square feet of ground are contained in it? Ans. 96X54=5144 square feet. 2. A lot of land, in form of a lony square, is 120 rods an length, and 60 rods wide; how many acres are in it? 120x60=7200 sqr. rods, then 400 =45 acres. Ans. 3. If a board or plank be 21 feet long, and 18 inches road; how many square feet are contained in it? 118 inches 1,5 feet, then 21x1,531,5 Ans. 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 boarıl be 8 inches wide, how much in leagth will make a square foot: RULE.—Divide 144 by the breadth, thus, 8)144 Ans. 18 ina 5. If a piece of land be 5 rods wide, low 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. 52x7=224 square inches, the Answer. 2. There is a triangular or three cornered lot of land whose base or longest side is 51rods; the perpendicular trom the corner opposite the base measures 44 rods; how niany 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 brise and perpendicular, by marking it down on a slate, paper, 19. 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 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 diam. eter is 4 feet i As 7 : 22 : : 4 : 12,57 the circumfe. rence. 2. What is the circumference of a circle whose diame Ler is 35 As 7 : 22 : : S5 : 110 Ans. and inversely as 22 : 7 : : 110 : 35, the diameter, &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, multipıy 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 inchos, 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 fer is 11 rods? 7854 By the second method, 11x11 = 121 Ans. 95,0334 rods. SECTION 2. OF SULIDS. Solids are estimated by the solid inch, solid foot, &c 28 of these inches, that is 12x12x12 make 1 eubic solid foot. ART. 6. To measure a Cabe. Definition... 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. EXAMPI.ES. 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 1,5X1,5X1,5=3,375 solid feet. Ans. • Or, 18x18x18=5832 solid inches, and 1=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 inust 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. 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 ? i ft. 6 in=1,5 foot. Prod. 1,125x9=10,125 solid feet, the Ans. in. in. in. solid in. 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 14., which will give the solid content m feet, &e. 2. A piece of timber being 16 inches broad, 11 inches thick, and 20 feet long, to find the content? Breadth 16 inches. Prod. 176x20=3520 then, 5520--144=24,4 feet, the Answer. 3. A piece of tinber 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, ins. 2. If a piece of timber be 18 inches broad and 14 in. ches 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 of tinber, 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 product will be the solid content. EXAMPLE. What is the solid content of a round stick of timber of equal biyness from end to end, whose diameter is 18 aches, and length 20 feet? |