What part of the rectangle is the triangle? What is the area of the rectangle? What, then, must be the area of the triangle? The area of the rectangle is 3 X 4 sq. units. The area of the triangle is of 3 X 4 sq. units. The area of a triangle is expressed by one-half of the product of the numbers that represent its base and altitude. 1. If the base of a triangle is 6 inches and the altitude 3 inches, what is its area? = of (36) sq. in. 9 sq. in. Find the area of the following triangles: 4. 2. Altitude 8 inches, base 6 inches. Cubic Measure. 352. Cubic Measure is used in measuring solids. 1728 cubic inches (cu. in.) = 1 cubic foot (cu. ft., = 1 cubic yard (cu. yd.). 27 cubic feet A solid that is 1 foot long, 1 foot wide, and 1 foot thick is called a cubic foot. Find a block of wood, a stone, or a box that is about as large as a cubic foot. Suppose Fig. 1 is drawn to a scale of 1 inch to 3 feet, or 1 yard. Then OD represents what length? What is the area of each face? a cubic yard? Does the figure represent A solid that is 1 yard long, 1 yard wide, and 1 yard thick is called a cubic yard. FIG. 3. FIG. 2. 354. Fig. 3 represents a cubic foot. If we cut off a piece an inch thick and divide it into inch cubes, how many of them shall we have? How many such pieces can be cut from a cubic foot? Then how many cubic inches in a cubic foot? A cubic yard is 3 ft. long, 3 ft. wide, and 3 ft. thick. Find the value of 3 X 3 X 3. Is the product the number of cubic feet in a cubic yard? What are the dimensions of a cubic foot? Find the value of 12 X 12 X 12, and compare it with the number of cubic inches in a cubic foot. Find how many cubic inches in a cube that measures 4 inches on each side. Build the cube with blocks (inch cubes). Volumes. 355. How many faces has a cube? Are they all equal? Is each face a rectangle? Figure 4 represents a solid 6 inches long, 4 inches wide, and 3 inches thick. How many faces has the solid? Is each face a rectangle? A solid bounded by 6 rectangles is called a rectangular solid. In Figure 4 each little cube represents 1 cubic inch. Find how many cubic inches there are. There are three layers, each containing 4 rows of the little cubes, as ABCD. Each row contains 6 little cubes, or cubic inches, hence the 4 rows contain 24 cubic inches. Since there are 24 cu. in. in each layer, in the 3 layers there are 3 X 24 cu. in., or 72 cu. in., which is the volume. Find the value of 6 X 4 X 3. Does the product equal the number of cubic inches in Fig. 4? (6 X 3 X 4) X 1 cu. in. cu. in. = The volume of a rectangular solid is expressed by the product of the numbers that represent its length, breadth and thickness. The number of cubic units (cu. in., cu. ft., cu. yd., etc.) in a solid is called its volume. 1. How many cubic inches in a brick 8 in. long, 4 in. wide, and 2 in. thick? 2. How many cubic feet in a wall 16 ft. long, 4 ft. high, and 2 ft. thick? 3. A hole is 3 ft. long, 2 ft. wide, and 6 ft. deep. How many cubic feet of earth have been removed? 4. A cistern is 6 ft. square and 9 ft. deep. How many cubic feet of water will it hold? How many cubic yards? 5. How many cubic feet in a cord? A pile of wood 8 ft. long, 4 ft. wide, and 4 ft. high is called a cord. 6. How many cords are there in a pile of wood 16 ft. long, 4 ft. wide, and 4 ft. high? 7. A cubic foot of water weighs 1000 oz. What is the weight of a cubic yard of water? 8. Find the dimensions of a box that will hold 24 cu. in. of water. 9. What are the width and depth of a cistern 6 ft. long that will hold 144 cu. ft. of water? 10. An ice box holds 20 cu. ft. of ice. It is 4 ft. long and 2 ft. deep. How wide is it? 356. What part of A is dark? Into how many parts is each strip divided in B? What part of each strip is the shaded square? What part of the large square? Then of one-tenth of B is one of B. When anything is divided into ten equal parts, what is each part called? When each of these ten parts is divided into ten parts, how many parts are there? When each of these hundred parts is divided into ten parts, how many parts are there? When anything is divided into tenths, hundredths, thousandths, etc., the parts are called decimal parts; that is, tenth-parts. 357. You have already learned that 10 hundredths make 1 tenth, and that 10 tenths make 1 one, just as 10 ones make 1 ten, 10 tens 1 hundred, and so on. You have also learned that tenths and hundredths may be written with the decimal point and without denominators. Fractions so written are called Decimals or Decimal Fractions. 358. Tenths must stand next to ones, because 10 of them make 1 one; hundredths next to tenths, because 10 |