365. The basis of the measurement of the circle and its parts is the relation of the circumference to the diameter.

366. By methods which are explained in more advanced works, it is ascertained that

The circumference of a circle is 3 times the length of the diameter; or, otherwise expressed, 22 times the diameter. The fraction expresses the relation still

more nearly.

367. As a closer approximation, the circumference of a circle may be stated as 3.14159+ times the diameter. The number 3.1416 is generally adopted, being sufficiently correct for all ordinary purposes; and it is usual to express this number by the Greek letter π (p).

П =

3.1416.

In the following formulæ, let C be the circumference, d the diameter, r the radius.

Note. It is manifest from Problem 62, p. 65, that a hexagon inscribed in a circle is six times the length of the radius of the circle, or three times the diameter, I which is double the radius; and it will be obvious from inspection, that the circle is a little longer than the perimeter of the hexagon-that is, a little more than three times the diameter.

Problem 15.

368. To find the circumference of a circle, the diameter or the radius being known.

Rule.-Multiply the diameter or double the radius

When the radius is the given quantity it must be doubled, whence the following formula

Note. The radius, multiplied by 3.1416, gives the half-circumference-i.e., an arc of 180°; whence, if the radius be 1, the half-circumference is 3·1416.

Problem 16.

369. To find the diameter or the radius, the circumference being known.

Rule. For the diameter divide the circumference by , for the radius divide the circumference by 2%.

From the above equations the formula for the diameter or for the radius is obtained

Instead of dividing by, we may multiply by its reciprocal, 3183.*

370. Exercises.

1. What is the circumference of a circle of which the diameter is 17 feet?

Ans. 53 4072 feet.

The number 3183 is the reciprocal of 3·1416—that is, 1 divided by 3.1416.

1

3.1416

= ⚫3183.

THE CIRCUMFERENCE, ETC.

117

2. Find the circumference of a circle of which the radius is 3 yards.

Ans.-21 yards, 2 feet, 11+ inches.

3. Find the length of a meridian circle (the earth's circumference taken through the poles), supposing the earth to be perfectly round, and 7912 miles in dia

meter.

Ans.-24,856 miles.

4. What is the length of the equator, supposing the earth's diameter there to be 7925 miles?

Ans.-24,898 miles.

5. Find the diameter of a circle of which the circumference is 75 feet.

Ans.-23 feet, 10+ inches.

6. Find the radius of a circle of which the circumference is 50 yards.

Ans.-7 yards, 2 feet, 10+ inches.

7. What is the diameter of the moon, supposing that she is perfectly round, and 6785.856 miles in circumference?

Ans.-2160 miles.

8. What is the radius of a circle the circumference of which is 314.16 yards?

Ans.-50 yards.

9. The diameter of a circle is 1024 feet; what is the circumference?

Ans.-4 furlongs, 34 poles, 5+ yards.

10. The circumference of a circle is 27 yards, 1 foot, 3 inches; what is the radius ?

Ans.-4 yards, 1 foot, 1+ inch.

Problem 17.

371. To find the area of a circle, the diameter being known.

Rule.-Multiply the square of the diameter by 7854. Formula

A

=

d2 x 7854.

Note 1.-This number, 7854, is one-fourth of 3·1416, and expresses the area of a circle whose diameter is 11 being also the square of the diameter. That is, if the diameter is 1 foot in length, the area is 7854 of 1 square foot; or, as it may be expressed roughly, a little more than three-fourths.

Note 2.-It is sometimes convenient to use the halfdiameter or radius. As the square of half a line is one-fourth of the square of the whole line (207), so

372. To find the diameter of a circle, the area being known.

Rule.-Divide the area by 7854, and extract the square root of the quotient.

1. Find the area of a circle, the diameter of which is 4 yards.

A = 42 x 7854.

A = 16 x 7854.

A=12.5664.-Ans.

The answer expresses square yards, and amounts to 12 sq. yards, 5 sq. feet, 14+ sq. inches.

AREA OF A CIRCLE.

119

2. Find the area of a circle, the diameter of which is 18 feet.

Ans.-28 sq. yards, 2 sq. feet, 67 + sq. inches.

3. What is the area of a circular field, of which the diameter is 379 links?

Ans.-1 acre, 0 roods, 20 poles, 15 + sq. yards.

4. Find the diameter of a circle, of which the area is 153.938 sq. feet.

Ans.-14 feet.

5. Find the diameter of a circle, of which the area is 1 acre.

Ans.-78 yards, very nearly.

6. Find the area of a circle, of which the radius is 5 yards.

Ans.-78 sq. yards, 4 sq. feet, 123 + sq. inches.

7. Find the diameter of a circle, the area of which is equal to 1 square foot.

Ans.-1.12838 foot; or 13.54 inches.

8. Find the area of a circle, of which the diameter is 5 feet.

Ans.-2 sq. yards, 5 sq. feet, 109 + sq. inches.

Problem 19.

374. To find the area of a circle when the radius and the circumference are known.

Rule. Take half the product of the radius by the circumference.

rC 2.

Note 1.—This rule may be observed to be similar to the first rule for finding the area of a regular polygon, as