Prob. 10. To find the Length of any Arc of a Circle.
Rule 1. As 180 is to the number of degrees in the arc,
So is 3 1416 times the radius, to its length.

Or as 3 is to the number of degrees in the arc,
So is 05236 times the radius, to its length.

Ex. 1. To find the length of an arc ADB (Prob. 8,) of 30 degrees, the radius being 9 feet.;

Rule 2. From 8 times the chord of half the arc subtract the chord of the whole arc, and of the remainder will be the length of the arc nearly.

Ex. 2. The chord AB (Prob. 8.) of the whole arc being 4-65874, and the chord AD of the half arc 2:34947; required the length of the arc.

2.34947
8

18.79576
4.65874

3) 14:13702

4.71234 auswer.

Prob. 11. To find the Area of a Circle, the diameter or circumference being given.

Rule 1. Multiply half the circumference by half the diameter. of the product of the whole circumference and diameter. Multiply the square of the diameter by 7854.

Or, take
Rule 2.
Rule 3.
Rule 4.

area.

Multiply the square of the circumference by '07958.
As 14 is to 11, so is the square of the diameter to the

Rule 5. As 88 is to 7, so is the square of the circumference to the

area.

Ex. To find the area of a circle whose diameter is 10, and circumference 314.159265

Prob. 12. To find the Area of the Sector of a Circle.

Rule 1. Multiply the radius, or half the diameter, by half the are of the sector, for the area. Or take of the product of the diameter and arc of the sector.

Note. The arc may be found by problem 10.

Rule 2. As 360 is to the degrees in the arc of the sector, so is the whole area of the circle, to the area of the sector..

Ex. What is the area of the sector CAB, the radius being 10, and the chord AB 16.

Prob. 13. To find the Area of a Segment of a Circle.

Rule. Find the area of the sector having the same arc with the segment, by the last problem.

Find the area of the triangle, formed by the chord of the segment and the two radii of the sector.

Then the sum of these two will be the answer when the segment is greater than a semicircle: but the difference will be the answer when it is less than a semicircle.

Er. Required the area of the segment ACBD, its chord AB being 12, and the radius EA or CE 10.

Prob. 14. To find the Area of a Circular Zone ADCBA. Rule 1. Find the areas of the two segments AEB, DEC, and their difference will be the zone ADCB.

Rule 2. To the area of the trapezoid DQP add the area of the small segment ADR; and double the sum for the area of the zone ADCB.

Prob. 15. To find the Area of a Circular Ring, or Space included between two Concentric Circles.

The difference between the two circles will be the ring. Or, multiply the sum of the diameters by their difference, and multiply the product by 7854 for the answer.

Ex. The diameters of the two concentric circles being AB 10

and DG 6, required the area of the ring contained between their circumferences AEBA, and BFGD.

Prob. 16. To measure long Irregular Figures.

Take the breadth in several places at equal distances. Add all the breadths together, and divide the sum by the number of them, for the mean breadth; which multiply by the length for the area.

Ex. The breadths of an irregular figure, at five equi-distant places being AD 8·1, mP 7·4, nq 9.2, or 101, BC 86; and the length AB 39; required the area.

8.1
7.4

9.2

10.1

8.6

5) 43.4

8.68

39

7812

2604

338.52 Ansr.

MENSURATION OF SOLIDS.

Prob. 1. To find the Solidity of a Cube.

Cube one of its sides for the contents; that is, multiply the side by itself, and that product by the side again.