VOLUMES OF RECTANGULAR SOLIDS Suppose that the figure below represents a rectangular solid 5 in. x 4 in. x 3 in. If it were cut into inch cubes, there would be one of these cubes for every square inch of the bottom layer, and as many similar layers as there are inches in the height. The square inches in the bottom are 5 x 4 = 20 sq. in. So there are 20 of the inch cubes in each layer. As there are 3 layers, there are 3 x 20, or 60 of these cubes in the solid. As the volume of an inch cube is 1 cu. in., the volume of 60 inch cubes is 60 cu. in. But 60 is the number obtained by multiplying 5 x 4 x 3. The volume of a rectangular solid is obtained by multiplying together its three dimensions expressed in units of the same denomination. 4. How many cubic yards of earth are removed for the foundation of a house 75 ft. by 54 ft., if the earth is removed to the depth of 21 ft.? 5. A cistern, in the shape of a rectangular solid, is 22 ft. by 14 ft. by 6 ft. How many gallons of water can it contain? 6. A bin is 8 ft. by 3 ft. by 6 ft. How many bushels can it hold? 7. In order to build a concrete wall, earth is removed to the depth of 6 ft. If the wall is 210 ft. long and 12 ft. wide, how many cubic yards of earth must be removed? 8. How many cubic yards of gravel are required to fill, to the depth of 6 in., a street 1 mi. long and 36 ft. wide? 9. How many cubical boxes 2 ft. each way would a storeroom 18 ft. by 12 ft. by 10 ft. hold? 10. The Sault Ste. Marie Canal is 1.6 mi. long, 160 ft. wide, and 25 ft. deep. Express in cubic yards the volume of water required to fill it. 11. A block of marble is 4 ft. by 3 ft. by 24 ft. How many tons does it weigh, if a cubic foot of marble weighs 170 lb. ? 12. How many pounds does a cedar beam weigh that is 14 in. by 10 in. by 40 ft., if a cubic foot of cedar wood weighs 38.1 lb. ? 13. A cubic foot of clay weighs 75 lb. Find, in tons, the weight of a clay bank 10 ft. by 4 ft. by 80 ft. 14. A box 9 in. by 8 in. by 6 in. is filled with mercury. Find its weight in pounds, if a cubic foot of mercury weighs 13,570 oz. 15. How many 3-in. cubes are required to fill a cubical box each of whose edges is 1 yd.? 16. A pile of 4-ft. wood 8 ft. long and 4 ft. high contains a cord. How many cords of wood in a pile of 4-ft. wood 120 ft. long and 12 ft. high? 17. Find the weight of the water covering an acre to the depth of 4 in. 1 cu. ft. of water weighs 1000 oz. A quadrilateral is a plane figure bounded by four straight lines. Figures 1 and 2 represent quadrilaterals. Parallel lines are lines which lie in the same plane and do not meet if extended in In the parallelogram ABCD, we may call AB the base, The side opposite the base of a paralleloDC is the upper or lower base. gram is sometimes called the upper base. base of parallelogram ABCD. tance between the bases. EB is the altitude of parallelogram ABCD. Draw a parallelogram ABCD, and, on the same base, AB, draw the rectangle ABEF. With scisThen fold on line sors cut out the whole figure ABCF. AD, and notice that the parallelogram is left after the removal of triangle AFD. Fold this triangle back into place, and then fold on line EB. Notice that the removal of triangle EBC leaves the rectangle. Then cut on the line AD and on the line EB. Place the two triangles together carefully and notice that they are equal. The parallelogram ABCD = ABCF — ADF. The rectangle ABEF = ABCF – BEC. Hence, the parallelogram ABCD = the rectangle ABEF. As the area of the rectangle ABEF is equal to the product of its base AB by its altitude EB, and the area of the parallelogram ABCD is equal to the area of the rectangle ABEF, then the area of the parallelogram ABCD is equal to the product of its base AB by its altitude EB. The area of a parallelogram is equal to the product of its base by its altitude. Thus, the area of a parallelogram, whose base is 12 ft. and altitude 5 ft. equals 12 x 5 60 sq. ft. = EXERCISE 66 Find the areas of the following parallelograms: 1. Base 10 ft. 6 in., altitude 6 ft. 4 in. 2. Base 17 ft. 3 in., altitude 9 ft. 8 in. 3. Base 12 ft. 6 in., altitude 8 ft. 3 in. 4. Base 15 ft. 5 in., altitude 9 ft. 4 in. 5. Base 36 ft. 9 in., altitude 8 ft. 4 in. 6. Base 40 ft. 3 in., altitude 7 ft. 6 in. 7. Base 27 ft. 9 in., altitude 8 ft. 7 in. 8. Base 28 ft. 4 in., altitude 6 ft. 3 in. A triangle is a plane figure bounded by three straight lines. ABC is a triangle. B A D FIG. 1. In the triangle ABC, we may call AC the base (Fig. 1). The altitude of a triangle is the perpendicular distance from the base to the opposite vertex. BD is the altitude of triangle ABC. In order to find the area of triangle ABC (Fig. 2), draw a line through C parallel to AB and a line through B parallel to AC, and draw the altitude BH. As ACDB is a parallelogram, its area is equal to the product of AC by BH. Cut on the line A BC. Place the triangle BCD on the triangle ABC. B H FIG. 2. It will be seen that the two triangles are equal. Therefore, the triangle ABC is equal to half of the parallelogram ABCD. Therefore, the area of the triangle ABC is equal to half the product of AC by BH. The area of a triangle is equal to half the product of its base by its altitude. Thus, the area of a triangle, whose base is 12 ft. and altitude 5 ft., equals one half of 12 times 5 = 30 sq. ft. EXERCISE 67 Find the areas of the following triangles: 7. Find the area of a right triangle whose base is 26 ft. and altitude 19 ft. (A right triangle is a triangle having a right angle.) |