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An angle is the space intercepted between two lines which intersect, or cross each other.
If one of the lines falls perpendicularly on the other, making an angle equal to 90 degrees, or the fourth part of the circumference of a circle, it is called a right angle. Thus the angle ABC, or the angle at B, in the above figure, is a right angle. If the space included between two lines be less than a right angle, it is called an acute angle; but if the space than a right angle, it is called an obtuse angle.
The rules for drawing the square, rectangle, rhombus, rhomboid, and trapezoid, are so simple, and so clearly illustrated by the figures, that it is deemed unnecessary to introduce them into this work.
T14. MENSURATION OF SUPERFICIES.
The square is a figure bounded by four right and equal lines, and contains four right angles.
To find the area of the square:—
Multiply one side into itself, or square the given side.
1. What is the area of a square whose side is 8 fourths of an inch? Ans. 4 of a square inch, or 4 square inches.
The above figure is 2 inches on each side, or 8 fourths of an inch; and it is divided into four squares, (each of which is 1 inch on each side,) and also into 64 small squares, each of which is of an inch on each side; and since the square of is, it is manifest that the above square contains of a square inch, or 4 square inches.
2. What is the area of a square 30 rods on each side? of one 40 rods on each side? of one 25 rods on each side? of one 15 rods? of one 12 rods? of one 10 rods? of one 6 rods? of one 80 rods? of one 120 rods? and of one 320 rods?
To solve the above by the sliding rule, set the slider as directed in 9, and over the number of rods on a side, found on the line D, will be found the number of square rods on the line C. Or, when the slider is in its usual position, that is, when the rule is shut, over the number of rods on a side of the given square, will be found the number of acres it contains on the C line. Thus, over 40 rods on the line D, at the right, we find 10 acres on the line C; and over 30 we find 5.6 acres ; over 25 we find 3.9 nearly; over 20, 2.5; over 15, 1.41; over 12, 0.9; over 10, 0.625; over 6, 0.225; over 80, 40; over 120, 90; and over 320, we find 640 acres.
A square piece of land 10 chains on a side contains 10 acres; if, therefore, the side of a square piece of land be given in chains, set 10 on D under 10 on C, and over the number of chains on a side, found on D, will be found the number of acres on the line C.
If the side of a square be given in inches, to find its area in feet by the sliding rule :-Place 1 on C over 12 on D, and over the given side, found on D, will be found the number of feet on the C line.
3. What will be the value of a plot of ground 15 rods square, at $5 the square rod? Ans. $1,125.
To find the value of a square piece of land at so much per rod, or chain :-Place the price of one rod, or one chain, found on C, over 1 on D, (calling the 10 one,) then over the side of the piece found on D, will be found the answer required, on C. 4. What is the value of a square field 17 chains on a side, at 25 cents per square chain? Ans. $72.25.
5. What would 15 feet square of plastering cost, at $0.015 per square foot? To find the side of a square, the area being given:the square root of the given area.
See ¶ 5 and 9.
To find the distance between the opposite corners of a square-Extract the square root of twice the area.
6. A square field contains 90 acres of land; I require the length of its side, and the distance between its opposite corAnswers, 120; and 169.7 rods.
In this example, before we extract the root by numbers, we must reduce the acres to rods; but it has been shown, that, when the slider is in its usual position, over any number of rods found on the line D, is the number of acres on the line C; and consequently, under the given number of acres found on C, will be found the number of rods on one side, viz. 120; and under twice the area, or 180 acres, will be found the distance between the opposite corners, viz. 169.7.
7. How many acres does a town contain which is 6 miles square? Ans. 23,040.
As 640 acres make one square mile, if we place 640 on the C line over 1 on the line D, then over 6 on D, will be found the number of acres on the line C.
Feet multiplied by feet produce feet.
Feet multiplied by inches, and the product divided by 12, give square feet.
Inches multiplied by inches, and the product divided by 144, give feet.
Feet multiplied by primes, or lines, and divided by 144, give square feet.
Inches multiplied by lines, or primes, and divided by 12, give square inches; or inches multiplied by primes, and the product divided by 1728, give square feet.
Primes multiplied by primes, and the product divided by 144, give square inches.-A prime is one-twelfth of an inch, and a square prime is of a square inch.
The rectangle is a figure bounded by four right lines, the opposite sides being equal and parallel, and its four angles right angles.
To find the area of a rectangle :
Multiply the length by the breadth.
1. What is the area of a board 8 inches long and 6 inches broad? Ans. 48 inches, or of a square foot.
2. What is the area of a piece of land 8 rods long and 5 Ans. 40 square rods.
The above plate represents such a piece of land; and the correctness of the rule may be demonstrated by counting the squares.
3. What is the area of a board 12 feet long and 10 inches broad? Ans. 10 feet.
Inches being twelfths of a foot, if we multiply the length in feet by the breadth in inches, the product will be twelfths of a square foot; and, consequently, to reduce the product to feet, divide it by 12. Or, (by the Rule of Three,) as 12 is to the length of the board in feet, so is its breadth in inches to its area in square feet. To find the area by the sliding rule, see [ 4.
4. What is the area of a board 13 feet in length and 15 inches in breadth ? Ans. 16.25 feet.
By the sliding rule, set the length in feet found on the line B, (viz. 13,) under 12 on the line A, calling the 13 thirteen feet, and the 12 twelve inches; then under the width of any board in inches found on the line A, will be found its area in square feet on the line B, its length being 13 feet.
5. What are the contents, or areas, of the following boards, each being 14 feet in length, and one of them 14 inches, one 18, one 20, one 22, one 30, one 10, one 8, one 6, and one 4 inches in breadth? Answers, 16.3; 21; 23; 25; 35;
11; 9; 7; 4 feet.
Having set 14 on B under 12 on A, under the respective widths found on A will be found the areas on the line B.
6. How many square feet are there in the four sides of a room, 22 feet long, 17 broad, and 11 feet in height? And how much would it cost to paper the walls at three cents for a square foot? Answers, 858 feet; $25.74.
7. I desire to cut off one square foot from a board 30 inches broad; how far from the end of the board must I saw it off? 144 30 Ans. 4.8 inches.
8. A piece of cloth 36 yards long contains 63 square yards; Ans. 12 yards.
I require its breadth.