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OF THE CIRCLE.

A

CIRCLE may be confidered as a regular polygon of an in finite number of fides.

The area of a circle is equal to the area of a triangle, whofe bafe is equal to the circumference, and height the radius.

The proportion of the diameter to the circumference may be found thus: Defcribe a polygon of a great number of fides, about a circle of a known diameter, and infcribe another of the like number of fides; find the perimeter of each, and the square root of their product will be the circumference. Hence the circumference of a circle whofe diameter is 1, is 3.141592653 58979323846264338327950288, of which number 3.1416 may be used, it being fufficiently accurate for most practical pur poses.

PROBLEM XIV. Plate 6. fig. 82.

The diameter being given to find the circumference

RULE.

Multiply 3.1416 by the given diameter; the product will be the circumference.

EXAMPLE I.

Required the circumference of a circle, whofe diameter is

14 feet.

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Ex. 2. The distance between the cogs of a mill-wheel, and the centre of the axle, is 5 feet; how many cogs, at 4 inches pitch, will the wheel admit of?

Ex. 3. The fame thing being given,

for 61 cogs.

Anf. 94.248 cogs. required the true pitch

Anf. 6 inches 2" 2" nearly.

Ex. 4. Suppofitg the earth to be an exact fphere, required its circumference, the diameter being 7958 miles.

Anf. 25000.8528 miles. Ex. 5. Required the circumference of a circle, whofe radius is 15 feet. Anf. 944 feet. Ex 6. What is the circumference of a circle, whofe diame ter is 48 inches ?`

Anf. 12 feet 8 inches 4 parts.

PROBLEM XIII.

The circumference of a circle being given, to find the diameter.

RULE.

Divide the circumference by 3.1416, and the quotient will be the diameter.

EXAMPLE I.

Required the diameter of a circle whofe circumference is

9448.

3.1416

3.1416)94.2480(30 feet
94248

Ex. 2. What is the diameter of a circle whofe circumference is 80 feet?

Anf. 25.464 feet. Ex. 3. Required the diameter of a circle, whofe circumfer

ence is 1000.

Ex. 4. Required the diameter of a circle, ference is 64 feet.

Ex. 5. Required the diameter of a milftone,

ference is 22 feet.

Anf. 318309. whose circumAnf. 20.368. whofe circumAnf. 7.0023

PROBLEM XIV.

The diameter and sircumference of a circle being given, to find

the area.

RULL.

Multiply half the radius into the circumference, and the product is the area.

EXAMPLE I

Required the area of a circle whofe diameter is 1

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CIRCLES are fimilar figures; and similar surfaces are as the fquares of their correfponding fides; therefore, as 12 is to .7854 (the area of a circle whofe diameter is 1) fo is the fquare. of any given diameter to the area required. Hence,

RULE II.

Multiply the fquare of the diameter by the common number 7854, and the product is the area.

EXAMPLE H.

Required the area of a circle, whofe diameter is 12 feet.

Anf. 113.0976 fquare feet. Ex. 3. How many fquare feet are in a table, whofe diameter is 6 feet? Anf. 28.2744 feet. Ex. 4. What is the area of a circular court, whofe diameter is 24 yards? Anf. 452.3904 Square yards. Ex. 5. How many fquare miles are in a great circle of the earth, its diameter being 7957 miles?

Anf. 49736071.58880750. Ex. 6. What is the area of a circle, whose diameter is 3 feet? Anf. 11.0446875 fquare feet. Ex. 7. Required the area of a circular garden, whose radius is 160 links of the English chain. Anf. 3 roods & poles 20 yds. Ex. 8. What is the area of a circle, whose diameter is 2 feet? Anf. 3.1416.

Ex. 9. Required the area of the ring between the circumferences of two concentric circles, their diameters being 20 and 35 inches. Anf. 137.445 fq. inches.

PROBLEM

PROBLEM XV.

The circumference of a circle being given, to find the area.

RULE.

Multiply the fquare of the circumference by .0795775, and the product will be the area.

EXAMPLE I.

Required the area of a circle whofe circumference is 1.

=

.0795775

I

.0795775

Ex. 2. What is the area of a circle, whofe circumference is 5 feet?

Anf. 1.9894375

Ex. 3. Required the area of a circle whofe circumference is too yards. Anf. 795-775Ex. 4. The expence of inclosing a circular court at 8s. per yard, amounted to 3201; required the

at 6d. per fquare yard.

expence of paving it, Anf. 12731. 58. 7d.

Ex. 5. How many fquare feet are in a circular table, whose

circumference is 25.1328 feet?

Ex. 6. Required the area of a circle,

is 31.416.

PROBLEM XVI.

Anf. 50.2656 fq. feet.

whofe circumference

Anf. 78.54.

The area of a circle being given, to find the circumference.

RULE I.

Divide the area by .0795775, and the fquare root of the quo

tjent will give the circumference,

RULE

1

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