A "circle" is a plane figure bounded by one continuous curved line, which at all points is a uniform distance from a point in the center of the figure. The distance around a circle is its "circumference" just as the distance around a square or oblong is its "perimeter." A straight line drawn from one point in the circumference to another point in the circumference through the center is a "diameter." A diameter divides a circle into two halves, each of which is called a "semi-circle," "semi" meaning half. Two diameters drawn at right angles to each other divide a circle into four quarters, each quarter being called a "quadrant.” A straight line drawn from the center to a point in the circumference is called a "radius," and several such lines are called "radii." Any part of the circumference of a circle is called an "arc," and the figure bounded by two radii and an arc is called a "sector" of the circle. Exercise 35-Oral. Go to the board to show all of these: 1. Be ready to draw a circle. 2. Draw the diameter or "through line" of this circle and say how long it is. 3. Into how many parts is the circle now divided? What is each of these parts called? 4. Draw another diameter at right angles to the first diameter. How long is this second diameter? Into how many parts is the circle now divided? 5. Rule two radii so that the ruler will show a distance of 1 inch between the points where the two radii touch the circumference. What is the length of each of these radii? 6. What is the part of the circumference between the two radii called? What is the remainder of the circumference called? 7. What is the figure bounded by the two radii and that part of the circumference which lies between the two radii, called? 8. What is a circle? What is half a circle called? What is a quarter of a circle called? 9. What is the curved boundary of a circle called? 10. What is a line drawn through the center, touching the circumference at two opposite points, called? Such a line divides the circle into what? 11. What is a line drawn from the center to the circumference of a circle called? Give one word. for many of them. 12. What is any part of the circumference of a circle called? 13. What is the figure bounded by two radii and an arc called? 14. How many sides has it? LESSON 17 The Ratio of the Circumference to the Diameter Measure the circumference and diameter of a coin, a wheel, a plate, and a spool. Arrange them in order. Compare each C with its D. (Coin) C: D= ? (Wheel) C: D= ? (Plate) C: D= ? (Spool) C: D= ? Write the ratio for each of these. Is the ratio about the same each time? What do we know is the relation of C to D? π The ratio shown by your comparison is a little over 3, or nearly 3.1416; therefore, for all ordinary purposes, 3.1416 or 34 will be used. This ratio is expressed by the Greek letter which is called "pi" (pronounced pi). By using the letter C to represent the circumference, D for diameter, R for radius, and for pi or 3.1416, we can show that the circumference equals the diameter multiplied by pi in this manner: C = DX T. Making a statement in this form is called an "equation." An equation is a statement showing the equality of two quantities by placing one before and one after an equality sign. Exercise 36—Oral. 1. Express the ratio 3.1416 as a common fraction (approximately). 2. If you have D, what must you do to get C? 3. If you have C, how would you find D? 4. If the diameter of the driving wheel on a locomotive is 10 feet, tell how to find the circumference of the wheel. How many feet would this wheel travel on the track in making one revolution? 5. Express the relation of D to R. 6. How can we find the radius when the diameter is known? 7. How can we find the radius when the circumference is known? 8. How can we find the circumference when the radius is known? 9. If C stands for circumference, D for diameter, R for radius, and for pi, read the following equations: This is a very short way of telling your rules. 10. Using equations, make statements to show: (a) That the diameter equals twice the radius; (b) That the diameter equals the circumference divided by pi; (c) That the circumference equals the diameter multiplied by pi; (d) That the circumference equals twice the radius times pi; (e) That the radius equals one-half the diameter; (f) That the radius equals one-half of the quotient of the circumference divided by pi; Exercise 37-(a) Oral. Tell how to work each of the following examples: (b) Written. Find the circumference, diameter, or radius as required: 7. Find the circumference of a table which has a diameter of 1 yd. 8. Find the radius of a circular lake which has a diameter of 100 yd. 9. Find the diameter of a circle which has a circumference of 9 yd. Carry it to 2 decimal places. 10. Find the circumference of a circle which has a radius of 1 yd. 2 ft. 3 in. Exercise 38—Written. 1. Find the circumference of Tom's bicycle wheel if the diameter is 28". (Use 34 for pi.) 2. The spokes in a certain wheel are 2 in. apart at the rim; what is the diameter of the wheel if there are 45 spokes? (Use 3.1416) |