C Trapezoid F B В G H The figure ABCD is a trapezoid. The parallel sides are AB and CD. The length of a trape E zoid is represented by a line parallel to and midway between the parallel sides, as EF. 7. Draw on paper a trapezoid ABCD of any convenient size, in which AB and CD are the parallel sides. Draw EF connecting the D middle points of AD and BC. Measure off on AB E the distance AH equal to EF. A A B В Draw HF. Cut out the trapezoid from the paper, and cut off the triangle HBF, and place it in the position HCF. You have now changed H the trapezoid into parallelogram. The two YF parallel sides of the trapezoid have been made the equal sides of the parallelogram, and one half the sum of the parallel sides is equal to AH, which is the base of the parallelogram. The altitude DG remains unchanged. As the area of a parallelogram is the product of its altitude and base, so the area of a trapezoid is the product of its altitude and one half the sum of its bases. D a A G Ή LESSON 114 1. Find the area of a trapezoid whose parallel sides are 70 ft. and 150 ft., and altitude 40 ft. 2. I have a flower bed in the shape of a trapezoid. The two parallel sides are 12 ft. and 14 ft., and the perpendicular distance between the parallel sides is 10 ft. Find the area. 3. The sum of the parallel sides of a trapezoid is 150 yards, and the perpendicular 75 yards. How many square yards are there in the area ? 4. One parallel side of a field in the shape of a trapezoid is 150 rd., the other 200 rd. How many square rods are there in the field, the perpendicular distance between the sides being 50 rd.? How many acres ? Make a diagram on a scale of 50 rd. to an inch. 5. Find the cost of cementing a cellar bottom 54 ft. , long and 18) ft. wide, at 623 ¢ a square yard. Since the area of a parallelogram is equal to the product of its length and breadth, either side will equal the area divided by the other side. 6. A blackboard has a surface of 105 sq. ft. What is the width of the board if it is 30 ft. long? 7. What is the length of a trapezoid whose parallel sides are respectively 24 ft. and 32 ft.? 8. How many feet are there in the perimeter of an equilateral triangle, each side of which is 51 yards long ? 9. How many strips of carpet, å yd. wide, will be needed for a room 18 ft. wide ? 10. If the above room is 22 ft. long, how many yards of carpet will be needed to cover the floor ? LESSON 115 1. If the base of a triangle is 12 yards and the altitude 8 yards, what is the area of the triangle ? 2. Draw a diagram of a right-angled triangle whose base is 12 rd. and altitude 6 rd., drawing to a scale of in. to a rod. Find its area. 3. Compare the area of the above triangle with the area of a rectangle 12 rd. long and 6 rd. wide. 4. The length of a rectangle is 80 rd. The length of its perimeter 220 rd. Find the width of the rectangle, and its area. Draw a diagram on a scale of 1 in. to 20 rd. 5. How many square yards of paper will be needed to cover the walls and ceiling of a room 18 ft. long, 15 ft. wide, and 12 ft. high? 6. At 3¢ a square foot, how much will it cost to sod a lawn 2 rods square ? 7. Draw a diagram representing a triangle whose area is 24 sq. in. 8. Draw a diagram representing a triangle whose area is 1 acre. 9. Draw a rectangle representing an area of 2 acres. 10. How wide must a board 16 ft. long be to contain 12 sq. ft.? 11. How much will it cost, at 12¢ a square foot, to lay a sidewalk, 8 ft. wide, around a rectangular plot of ground 1000 ft. by 600 ft. ? 12. Show by a diagram the difference between 3 inches square and 3 square inches. How many square feet of land are there in a buildin lot 25 ft. wide at one end, 20 ft. at the other, and 200 ft. deep? LESSON 116 А B CIRCLE A Circle is a plane figure bounded by a curved line, called its Circumference, every part of which is equally distant from a point within, called the center. The Diameter of a circle is a straight line drawn from any point in the circumference, through the center, and terminating in the circumference opposite, as AB. The Radius of a circle is a straight line drawn from the center to the circumference, as CD. 1. Draw on the blackboard a circle having a diameter 7 in. long. With the aid of a string, or tape line, get the length of the circumference of your circle. Divide the circumference by the diameter. Do you find that the circumference is about 34 times the diameter ? 2. Can you tell how to find the circumference when the diameter is given ? 3. Find the circumference when the diameter is 8 inches; 12 inches ; 24 feet; 20 yards ; 12 rods. 4. Find the circumference when the radius is 3 inches; 5 inches ; 21 yards ; 1.75 rods. 5. If the circumference equals the diameter multiplied by 31, how would you find the diameter when the circumference is given ? 6. Find the diameter when the circumference is 44 inches ; 66 feet; 90.2 yards; 11,044 rods. 7. Find the circumference of a circle whose diam. is 1 rd. 8. What is the diameter of a circle whose circumference is 18% feet? LESSON 117 Area of a Circle. The area of a circle can be found by multiplying the circumference by one half of the length of the radius. According to this rule, we regard the area of the circle as equal to the sum of the areas of a number of equal triangles. If there were only eight of these triangles, as in Fig. 1, it is evident there would be considerable difference between the sum of the areas of the triangles and that of the circle. But if the number of triangles were increased to thirty-two, as in Fig. 2, the sum of the areas of the triangles would approach much nearer the area of the circle. If the number of triangles were still further increased, they would form a plane figure that could hardly be distinguished from the circle. Now the sum of the areas of the triangles can be found by multiplying the sum of their bases by one half of their altitude. Therefore the area of a circle can be found by multiplying its circumference by one half of its radius. Suppose the radius of a circle to be 3 inches. 189 in. And the area = 189 * 1 of 3 18% x = 284 sq. in. 1. Find the areas of the following circles when the radius is 3 in.; 5 in.; 8 in.; 5.5 feet; 6.25 rods. 2. Find the areas of the following circles when the diameter is 10 in.; 14 ft.; 28 yd.; 16.8 rd. 1 |