MENSURATION 3735 



MENSURATION 



(1) Length = 8 ft. 

 Width = 12 ft. 



Area In sq. ft. = 8X 12 = 96 



(2) Length = 14 mi. 

 Width = 20 mi. 



Area in sq. mi. = 14X 20 = 280 



(3) Length = 121/2 yd. 

 Width = 6 yd. 



Area in sq. yd. = 6x 12% = 75 



In school or at home the child learning area 

 should do much real measuring of surfaces; 

 should draw to represent these surfaces, and 

 divide into square units square inches, square 

 feet and square yards to realize that the sur- 

 face of the rectangle is as many square units 

 as the product of the numbers denoting the 

 length and the width (study the article REC- 

 TANGLE in connection with this subject). 



In general, to find areas of other plane sur- 

 faces we compare them with the rectangle. 



The Parallelogram. A parallelogram is a 

 plane figure having its opposite sides parallel. 

 The rectangle is a parallelogram^ but there are 

 other parallelograms whose angles are not right 

 angles (see Fig. 2). 

 & 



FIG. 2 



6 



The base of this parallelogram is 12 units, 

 and its altitude or width is 4 units (note heavy 

 dotted line). See how this may be made into 

 a rectangle by cutting off a b c, and fitting it 

 on at e j g. Make a drawing, and do this. It 

 is now a rectangle whose length is 12 units and 

 whose width is 4 units, and so has an area of 

 48 square units. 



The area of a parallelogram is the same as the 

 area of a rectangle of the same length and width. 

 (Note the width is the perpendicular distance be- 

 tween the sides.) 



Triangle. A figure bounded by three straight 

 .lines is a triangle (see Fig. 3, I and II). 



i n 



FIG. 3 



Examine Fig. 3, I. See that it is half the 

 rectangle in which it is drawn. Now study II, 



and see how it could be made into a rectangle. 

 The area of I is one-half the area of the rec- 



\ V P\ 



tangle, or ^- squares, or 12% squares. The 



area of II is equal to the area of a rectangle 

 whose base is 5 and altitude 5, that is, whose 

 base is one-half the base of the triangle, and 

 whose altitude is the same as that of the 

 triangle. In, other words, its area is one-half 

 the area of the rectangle in which it is drawn. 

 Draw several triangles and by cutting and 

 arranging see how each compares with the area 

 of a rectangle of the same base and altitude. 



The area of a triangle equals one-half the area 

 of a rectangle of the same dimensions; or, The 

 area of a triangle equals one-half the product of 

 its base and altitude. This rule may te ex- 

 pressed in the three following mathematical state- 

 ments : 



Area of triangle = 



Base x Altitude 



Area of triangle = BXY 



T5 



Area of triangle-=Ax y 



How many triangular samples of cloth, each 

 having a 6-inch base and 4-inch altitude, can 

 be cut from a piece of cloth 18 inches long 

 and 4 inches wide? 



Show this by a drawing and cutting. Can you 

 draw and cut in another way? (Study the 

 article TRIANGLE in connection with this sub- 

 ject.) 



Trapezoid. A trapezoid has two of its sides 

 parallel (called the bases) and two sides that 

 are not parallel (see Fig. 4, I and II). The 



altitude is the distance between the bases, as 

 shown by the heavy dotted lines in each figure. 

 Examine I and see how it can be made into a 

 rectancle 13 inches long and 4 inches wide. 

 Draw and cut to show this. Look at II and 

 see that it can be made into a rectangle 13 

 inches long and 4 inches wide. 



