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9. How many panes of glass, each 8 by 10 inches, will be required for a window 5 feet in height, and 2 feet 8 inches broad? Ans. 24.
10. There is a house with 3 tiers of windows, and 3 windows in a tier; the height of the first tier is 7 feet 10 inches, the second is 6 feet 8 inches, and the third 5 feet 4 inches; and the breadth of each is 3 feet 11 inches; what will the glazing come to at 183 cents per square foot?
11. The circumference, or perimeter of a room is 90 feet 11 inches, and its height 9 feet 8 inches; what is the area of its walls in square yards?
Ans. 97 yards, 5.87 feet, nearly.
To find the side of a square whose area shall equal that of a given rectangle :
Extract the square root of the area of the rectangle.
Or, when the length and breadth of the rectangle are given, to find the side of an equal square by the sliding rule:
Set one side of the given rectangle, found on the Cline, over the same number found on the line D; then find the other side on the C line, and under it will be found the side of an equal square on the line D.
1. The sides of a rectangle are 40 and 10; I require the side of an equal square.
Set 10 on C over 10 on D, and under 40 found on C, will be found 20, the side of the equal square, on the D line. 2. What is the side of a square equal to a rectangle 25 by 9 feet? Ans. 15 feet. 3. What is the side of a square equal to a rectangle 45 by 80 rods? Ans. 60 rods.
Having found the side of an equal square by the sliding rule, the area of the rectangle may be found in acres, or feet,
by the rules given in 14. Thus, in the above example, having found the side of the square, 60 rods, shut the slider, and over 60 rods found on D, we find the area, viz. 22.5 acres.
4. What is the side of a square equal to a rectangle whose sides are 32 by 24 rods? And what is its area?
Answers, 27.75 rods, and 4.8 acres.
5. The length of a rectangular pavement is 47 feet 9 inches, and its breadth 13 feet 11 inches; how many stones, each 17 inches broad, will serve to pave it ?
inches long and 10
6. The length of a house is 40 feet 9 inches, and the sloping height of the roof above the walls is 19 feet 5 inches; how many slates will cover the roof, supposing each slate to cover 17 square inches? Ans. 1302.
The rhombus is a figure bounded by four right and equal lines, and has two of its angles obtuse, and two acute, the opposite angles being equal, and its opposite sides parallel.
The area of a rhombus is equal to that of a rectangle, one side of which is equal to one side of the rhombus, and the other equal to its perpendicular breadth; the rhombus abcd being equal to the rectangle abeg, and the triangle bec being equal to the triangle adg.
Therefore, to find the area of a rhombus :
Multiply the length of one side by its perpendicular breadth.
1. What is the area of a rhombus, one side of which is 6 chains and 20 links, and whose perpendicular breadth is 5 chains and 45 links? Ans. 3 acres, 1 rood, and 20 rods.
2. What is the area of a rhombus, whose side is 40, and perpendicular breadth 32 rods? Ans. 8 acres. By the sliding rule we find the side of an equal square to be 35.75 rods, and the area 8 acres.
3. How many square feet does a rhombus contain, whose side is 40 inches, and whose perpendicular breadth is 22.5 inches? Ans. 6.25 feet.
By the rule, we find the side of an equal square to be 30 inches; then placing 1 on C over 12 on D, (calling the 1 one foot and the 12 twelve inches,) and over 30 inches will be found the area in feet, viz. 6.25.
4. What is the area of a rhombus, whose side is 13 feet and perpendicular breadth 32 inches? Ans. 34 feet.
5. How many square yards does a rhombus contain, whose side is 37 feet 10 inches, and the perpendicular height 28 feet Ans. 120 yards, 7 feet, 102 inches.
The rhomboid is a figure bounded by four right lines, having its opposite sides equal and parallel, and two of its angles obtuse and two acute, the opposite angles being equal.
The area of the rhomboid is equal to that of a rectangle whose length is equal to the longest side of the rhomboid, and its breadth equal to the perpendicular breadth of the rhomboid.
This is evident from the figure, the triangle bec being equal to the triangle agd, and the rectangle abeg being equal to the rhomboid abcd.
Therefore, to find the area of a rhomboid :—
Multiply the longest side by its perpendicular breadth.
1. What is the area of a rhomboid, whose length is 37 feet, and perpendicular breadth 5 feet 3 inches?
Ans. 194.25 feet.
2. What is the area of a rhomboid 17 feet in length and 22 inches in perpendicular breadth ?
Ans. 31.2 feet, nearly.
The trapezoid is a figure bounded by four right lines, having two of its sides parallel, but of unequal length.
By inspecting the above figure, it will be seen, that thè trapezoid, abcd, is equal to a rectangle, one of whose sides is equal to half the sum of the two parallel sides of the trapezoid, and the other equal to the perpendicular distance between these two sides. For, since eb is equal to cf, and ah is equal to gd, it is evident that gf or he is equal to half the sum of ab+cd; and since the triangle ebn is equal to the triangle nfc, and the triangle ahs is equal to the triangle sgd, it is manifest that the rectangle hefg is equal to the trapezoid abcd. But hg is the perpendicular between the two parallel sides of the trapezoid; and, consequently,
To find the area of the trapezoid : -
Multiply half the sum of the parallel sides by the perpendicular distance between them.
1. The parallel sides of a trapezoid are 750 and 1,225 links, and the perpendicular distance between them is 1,540 links; what is its area? Ans. 15 acres, 33 rods.
2. How many square feet are there in a board 12.5 feet in length, and 15 inches wide at one end and 11 inches at the other, the ends being parallel? Ans. 13.54 feet.