CHAPTER XI AREAS AND VOLUMES OF SIMPLE FIGURES 76. Square Measure. One of the many applications for squares and cubes of numbers is in the measurement of area and volume. As most of us know, area is a measure of surface, or "flat space," and it is measured in terms of the square unit. Thus, a room has so many square feet of floor space, a roof requires a certain number of shingles per 100 sq. ft. of surface, etc. In this case the area is measured by the unit known as the "square foot." A square foot is equal to the amount of surface bounded by a square having each side 1 ft. in length. The base of a milling machine is 3 ft. wide and 5 ft. long, and the floor space covered by the base is 15 sq. ft., as shown in Fig. 5' FIG. 30.-Diagram of milling machine base. = 30. It can readily be seen that the area is also equal to the product of the length times the width, since 5 X 3 15. The same is true of any rectangle, that is, any figure bounded by four sides and all four corners of which are right angles or square corners. Rule. To find the area of a rectangle, multiply the length by the width. The grate in a firebox is 8 ft. long and 6 ft. wide. What is the area of grate surface? To get the area, we multiply the length by the width. It must be emphasized here that we are not multiplying just "8 X 6," but rather "8 ft. X 6 ft." We see at once that the numerical result is 48, but what becomes of the "feet?" Since we have multiplied feet by feet the answer must be the square of feet, or, as it is called, "square feet"; therefore, FIG. 31.-Square and triangle, each of which has an area of 1 square foot. = 8 ft. X 6 ft. 48 sq. ft., Answer. A square foot of area does not necessarily have to be in the shape of a square, but may have any shape as long as it contains. the required amount of area. Figure 31 shows a triangle and a square, each of which has an area of 1 sq. ft. FIG. 32.-Diagram of 1 foot square divided into square inches. The square inch is another common unit of area. This is much smaller than the square foot, being only one-twelfth as great each way. If a square foot is divided into square inches, it will be seen to contain 12 × 12, or 144 sq. in., Fig. 32. It will be readily seen that the area of any square is equal to the product of the side of the square by itself. In other words, the area of a square equals the side "squared" (referring to the process explained in Art. 67). Care must be taken to use the same units for the length and width. Thus, you cannot multiply inches by feet and call the product square inches or square feet. Both length and width must be in the same unit, that is, both in feet, or both in inches, etc. If the unit of length is feet, the product will be square feet; if inches, the product will be square inches. As mentioned before, 1 sq. ft. is the area of a square 1 ft. on each side, and if divided into square inches, will be found to contain 122, or 144 sq. in. Likewise, a square yard is 3 ft. on each side and, therefore, contains 32 = 9 sq. ft. The following table gives the relation between the units ordinarily used in measuring areas: Often we know the area of a rectangle and the length of one side and we wish to find the other side. This is another case of reversing the operation, or working the problem backward. Hence, we merely divide the area by the known side to obtain the unknown side. Example: A storage room has 1000 sq. ft. of floor space. Because of its contents it is only possible to measure the length, which is found to be 50 ft. What is the width? 1000 sq. ft. ÷ 50 ft. (Check.-50 ft. x 20 ft. = = 20 ft., Answer. In the case of a square, it is only necessary to know the area in order to find the length of the side. Since the area of a square equals the side squared, the length of the side is simply the square root of the area. 77. Area of a Circle.-If we take a circular piece of cardboard, divide it into triangular sections as in Fig. 33, and cut one of the small sections in two equal pieces, we can then place the sections together as in Fig. 34. The figure thus found is practically a rectangle having a length equal to one-half the circumference of the circle and a width equal to one-half the diameter. Hence, the length equals × 3.1416 × diameter, and the width equals X diameter. Since the area equals length times width, we have The area of this rectangle is also the area of the circle, because dividing the circle into parts does not change its area. Therefore, www LENGTH FIG. 34.-Triangular sections of circle placed together. the area of any circle is found by squaring the diameter and multiplying the square by .7854. From the formula, it is evident that the ratio of the areas of any two circles will equal the ratio of the squares of their diameters. Consider two circles of diameter d and D respectively. 2. In a certain pump the ratio of the piston area on the steam end to that on the water end is as 4 is to 2.25 and the diameter of the piston on the water end is 6 in. Find the diameter of the piston on the steam end. Finding the Diameter from the Area.-Very frequently the area of a circle is known and it is desired to find the diameter. Just as we worked backward in previous cases, so we can in this case. To find the area of a circle we first square the diameter, and second, multiply by .7854. To find the diameter from the area, we reverse the process and work backward from the last operation: First, divide by .7854. Second, extract the square root. |