The curved surface of the cylindrical object (Fig. 263) is bounded by circles. On one of the circles mark a point P. Place P on point A of a straight line AB, and roll the object, without sliding, so that the circle always touches the line, until point P touches line AB a second time, at C.

FIG. 263

The circle may then be measured by measuring AC. Record the result in the table.

Measure the diameters of the circles measured above, and record the lengths in the table.

For each object find the ratio of the length of the circle to the diameter. With careful measurements these ratios should all be the same.

Find the average of the ratios in the table.

135. Circumference. The length of a circle is called circumference. It has been proved by mathematics that for all circles the ratio of the circumference to the diameter is the same. The ratio, computed approximately to three figures, is 3.14, or 34. The exact value of the ratio is denoted by the Greek letter (read

pi). It stands for the letter p, and is the first letter in the Greek word for perimeter. It came into use as a literal number in the middle of the eighteenth century. The fact that the ratio of the circumference to the diameter is the same for all circles was known to mathematicians long before that time, but they assigned to it different values. In one of the earliest books, written by Ahmes about 1700 в.C., π is given equal to 256 or 3.1604. The Jews considered π equal to 3.

Denoting the circumference by c and the length of the diameter by d, we have the formula

Let r be the length of the radius of a circle. Show that c=2πr

Translate these two formulas into words.

EXERCISES

In the following exercises use π=3.14.

1. Mary wishes to make a circular lamp shade whose diameter should be 23 inches. How much fringe does she need if she allows 1 inch for waste in joining the ends?

Solution: This is really a problem of finding the circumference when the diameter is known.

2. Helen's father is making a circular flower bed 12 feet in diameter. He has sent Helen to buy the plants which are to be placed along the border 6 inches apart. How many plants are needed?

3. Find the length of the equator, assuming the diameter of the earth to be 8,000 miles approximately.

4. The diameter of a circular grass plot is 56 feet. Find the distance around it.

5. What is the circumference of the largest circular table top that can be cut from a square whose side is 36 inches long?

6. How much lace will be needed to make an edge for a doily 24 inches in diameter, if we allow inch for the seam and 3 inches for fulling in?

7. What radius should be used to mark out a circular flower bed if you have 48 plants which are to be placed 6 inches apart to form the border?

8. Find the diameter of a tree trunk at a height where the circumference is 74 inches.

Suggestion: Use the method of Exercise 7.

9. The circumference of a circle is 42 inches. Find the diameter and radius.

10. A circular pond is surrounded by a walk whose inner circumference is 98 feet. Find the diameter of the pond.

11. When ordering clothing from a mail-order house one must state the size. This is easily determined for some articles. The size

of a boy's coat is the number of inches of the chest measure. The size of a collar is the number of inches between the outer ends of the button holes. But the size of a shoe is not the length of the shoe, and clothes for a small boy or girl are usually ordered according to his or her age.

The size of a man's hat may be determined by measuring the distance around his head and dividing the result by 34.

If the distance around a man's head is 22 inches, what is the size of his hat?

12. Two athletes are running on a circular track. One is running 4 feet farther from the center than the other. How much

farther will he have to run? Work this out for various diameters, e.g., 10, 15, 18.

13. The diameter of a wagon wheel is 42 inches. What distance does the wagon move when the wheel makes one complete revolution?

14. Explain why the speedometer of an automobile does not register correctly when oversized tires are put on the wheels.

15. Find the circumference and diameter of a wheel which makes 700 revolutions going one mile.

Suggestion: Show that 700c=5280.

16. Find, in feet, the distance traveled by a point on the rim of a fly-wheel which is 18 inches in diameter and makes 500 revolutions a minute.

17. The rear wheel of a wagon is 4 feet in diameter, and the fore wheel 3 feet. How many more revolutions than the rear wheel does the fore wheel make when the wagon travels one mile?

18. Denoting the circumference of a wheel by c, the distance it travels by l, and the number of revolutions it makes by n, make a formula which expresses l in terms of n and c.

19. By means of the formula c=3.14d, find, to 3 figures, the value of c if d=1.26; 4.08; 2.39; 18.2.

20. By means of the formula c=2d, find the value of c if d=3; ; 2; 4.6; 133.

21. By means of the formula c=3.14 d, find the value of d if c=6.20; 8.42; 9.36.

GRAPHICAL REPRESENTATION OF THE FORMULA

c = πd

136. Relation between diameter and circumference. We have seen that by means of the relation c=3.14d, we are able to determine for any given value of d the corresponding value of c, i.e., that the value of c depends upon the value of d. In the table below write the values of c corresponding to the values of d given in the first row: