LESSON 40 1. With the aid of a string or compasses draw a circle. 2. Draw a diameter and two radii. 3. Write diameter, circumference, radius, arc, each in its proper place. 4. Measure as accurately as you can the length of the circumference and of the diameter, and divide the circumference by the diameter. Compare your quotient with 3.1416. The exact ratio of a circumference to its diameter cannot be accurately expressed in numbers. Mathematicians have agreed to denote it by the Greek letter T (pronounced pie). I is very nearly equal to 3.1416. If the diameter of a circle is 4 in., the circumference will be 47, or about 4 times 3.1416. Make the following calculations, considering the length of the circumference to be at times the diameter: 5. Diameter 2 in.; circumference = ? 6. Diameter = 4 in.; circumference = ? 7. Radius 3 in.; circumference = ? 8. Radius 21 in.; circumference = ? 9. Circumference= 6 in.; diameter = ? Radius = ? 10. Circumference= 9 in.; diameter = ? Radius = ? 11. Circumference = 12 in.; diameter = ? Radius = ? 12. Draw a figure to show that a circle may be regarded as composed of a very large number of triangles, the sum of whose bases is the circumference of the circle, and the radius of the circle their common altitude. How may 13. How do you find the area of a triangle ? the rule for finding the area of a triangle be applied in finding the area of a circle ? 14. Find the area of a circle whose diameter is 4 inches and circumference 12.5664 inches. LESSON 41 Make the following calculations : 1. Radius 4 in. ; diameter ? Circumference ? 2. Diameter = 6 in. ; radius ? Circumference ? Area = ? 3. Diameter 8 in. ; radius ? Circumference ? Area ? 4. Circumference 12 ft. ; diameter = ? Radius ? Area = ? 5. Circumference = 30 yd. ; diameter 30 yd. ; diameter = ? Radius = ? Area Find the diameter, radius, circumference, and area of each of the following circles, drawn on a scale of 1 in. to 24 feet : 6. 7. 8. 9. = ? O 10. Draw on paper a rhomboid whose sides are 4 in. and 3 in., and altitude 21 in. Find its area. 11. Cut the rhomboid out from the paper, and change it into a rectangle, keeping the same base and altitude. Find area, and compare it with the area of the rhomboid. 12. Draw on paper a trapezoid, making its parallel sides 3 in. and 4 in., and altitude 2 in. Find its area. 13. Cut the trapezoid out from the paper, and change it into a rectangle. Find area, and compare it with the area of the trapezoid. 14. Compute the area of the trapezium at the right, drawn on a scale of 4 of an inch to 10 rods. 15. Find the value, at $50 an acre, of a piece of land having the shape of a trapezium, the diagonal of which is 40 rd., and the perpendiculars from this diagonal to the opposite corners 16 rd. and 18 rd. respectively. 16. Draw two trapeziums of different shapes, making the diagonal of each 3 in., and the perpendiculars from this diagonal to the opposite corners 2 in. and 11 in. respectively. Find the area of each. VOLUMES LESSON 42 A Solid is that which has three dimensions, — length, breadth, and thickness or height. The Volume of a solid is the quantity of space it occupies, and is expressed by the number of times it contains a cubic unit used as a measure. CUBIC UNIT How many 1. Name three cubic unit measures. Describe each. 2. In what particular respect are they alike? Wherein do they differ? 3. How many sides has each ? edges ? How many corners ? 4. What is the volume of a solid one inch long, one inch wide, and one inch high? 5. What is the volume of a solid two inches long, one inch wide, and one inch high? 6. What is the volume of a solid two inches long, two inches wide, and one inch high? 7. What is the volume of a solid two inches long, two inches wide, and two inches high? How many cubic inches in each layer? How many layers ? A Prism is a solid whose sides are parallelograms, and whose ends or bases are equal polygons parallel to each other. A prism is named from the form of its base as triangular, square, quadrangular, pentagonal, etc. 8. Find the volume of a quadrangular prism whose bases are 2-inch squares and height 4 inches. 9. How many cubic inches in each layer? How many layers are there? 10. Can you give a rule for calculating the volume of a prism when you know its dimensions ? 11. Is it true that the volume of a prism equals the area of the base multiplied by the altitude ? 12. What is the volume of a cube whose edge is 5 inches ? LESSON 43 1. What is the volume of a prism whose base is 4 ft. by 3 ft., and height 6 ft. ? 2. How many cubic feet in a pile of wood 8 ft. long, 4 ft. wide, and 4 ft. high ? Such a pile is called a cord. |