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MENSURATION FOR BEGINNERS.
MENSURATION OF PLANE SURFACES.
PARALLELOGRAMS AND TRIANGLES.
DEFINITIONS. 1. Mensuration is the art of finding the lengths, areas, and volumes of bodies.
2. The Mensuration of Surfaces enables us to measure the lengths of lines and the areas of surfaces.
3. A Surface, or Superficies, is the upper face or outside of a body, such as the top of a table.
4. A Plane Surface is one which is perfectly even or flat, like the surface of a lake.
5. The Area of a surface is the space contained within its boundaries; thus the area of a field is the quantity of land contained in it.
6. The boundaries of a surface are lines.
7. When lines intersect or cross each other, the place where they intersect is called a point.
8. The length and breadth of a surface are called its dimensions.
Lines have only one dimension ; viz., length.
The lengths of lines are expressed in Linear Measure, areas in Square Measure.
9. An Angle is the opening between two straight lines meeting each other in a point.
Thus the lines A and B meet in the point C, and form the angle C or AC B. When
Angle. an angle is denoted by three letters, the middle letter is always at the angular point.
10. Right Angle. When a straight line C D, standing on another straight line AB, makes the angle C D A equal to the angle 0 DB, each of these angles is called a right angle, and the straight line CD, standing on the other A B, is Ā said to be drawn at right angles
Right Angle. to AB. CD is also said to be perpendicular to A B.
11. Parallel straight lines are such as are the same distance from each other through. out their entire length, like the Parallel Lines. lines of a railway.
12. The boundaries of a figure are called its sides.
13. A Parallelogram is a four-sided figure which has its opposite sides equal and parallel.
Parallelograms are of four kinds—the Square, Rectangle, Rhombus, and Rhomboid.
The Square and Rectangle are also called Rectangular Parallelograms; the Rhombus and Rhomboid, Oblique Parallelograms.
THE SQUARE 14. A Square is a parallelogram which has all its sides equal, and all its angles right angles.
15. To find the area of a square when the side is given in one denomination only.
RULE. Multiply the length of the side by itself, and the produet will be the area.
Sappose we have a square, ABCD, of which each side measures 3 inches.
Draw straight lines, an inch apart, parallel to the sides. The square is thus divided into 9 equal figures, each of which is a square an inch long ind an inch broad. Such a square Ā is called a square inch; and the
Square. whole figure containing 9 of them, the area of the square is said to be 9 square inches.
The number of square units in each horizontal row will obviously be equal to the number of linear inches in the side A B; and the number of these rows will be equal to the number of linear inches in the side AD; therefore the number of square inches in the whole rectangle will be the number of inches in A B multiplied by the number of inches in AD.
Ex. 1. Required the area of a square floor of which each side measures 12-5 yds.
Area of floor=12.5 x 12.5=156.25 sq. yds. Ex. 2. Find the area of a square table whose side is 37 yds. Area=34 x 34=***= 488=1046 sq. yds.
Ex. I. Find the area of the squares whose sides are respectively:(1) 42 yds., 25 ft., 218 in., 625 yds., 305 ft., 41 in., 19 ft. (2) 30.25 yds., 1.425 yds., 6.75 in., 24.5 in., 14.4 ft., 8.625 ft. (3) 67 yds., 27 yds., 71t yds., 18} ft., 425 in., 152 ft., 20in. (4) Required the area of a square whose side is 29 yds.
(5) The length of a side of a square lawn is 36 yds.; what is its area?
(6) How many yards are there in a square bowling-green whose side measures 47.25 yds. ?
(7) Find the area of a square foor whose side is 259 ft. (8) How many square feet are there in 17 ft. square ? [Observe the difference between the expressions “feet square and square feet.” To say that any space measures 17 ft. square, implies that each side of it is 17 ft. Iong, and consequently that it contains 17 x 17 or 289 sq. ft; while the words 17 sq. ft. refer to a figure whose area con. tains 17 square feet. ]
(9) Find the difference between 15 sq. ft. and 15 ft. square. (10) A square field is 625 yds. long, how many acres are there in it? [Reduce the sq. yds. to acres. 4810 sq. yds.=1 acre.]
(11) A square field is 193 yds. long, how many acres are there in it?
(12) Find the area of a square whose perimeter is 256 yds. [The perimeter of a square is the distance round it. As all the sides are equal, divide the perimeter by 4, and the quotient will give the length of the side, from which the area may be found.]
(13) Required the area of a plot of land whose perimeter is 18.4 yds.
(14) If 176 yds. of fencing are required to enclose a square field, what is its area ?
16. To find the area of a square frame, or of a path running round a square plot of ground; i.e., to find the difference between the areas of two squares,
RULE. Find the area of each square, and subtract the area of the inner square from the area of the outer square.
17. When a walk of uniform width runs round a square plot, EFGH, it is evident that the figure ABCD, including the plot and the walk, is also a square.
18. If E F, the length of the inner square, be given, we must add twice the width of the walk to E F, in order to obtain the length À B of the outer square. But, if A B, tho levgth of the outer squaro, be given, we must sul,