To Find The Length Of Any Arc Of A Circle

Rule. - From eight times the chord of half the arc, subtract the chord of the whole arc, and one-third of the remainder will be the length nearly.

Required the length of the arc ABC, fig 9. the chord AB of half the arc being 8 1/2 ft., and chord AC of the whole arc 16 It. 8 inches.

41.334

8.5 X 8 = 68.0 and 68 0 - 16.666 = ---------= 13.778 c. feet the length of the arc.

Fig. 9.

To Find The Area Of The Sector Of A Circle

Rule. - Multiply the length of the arc by half the length of the radius.

The length of the arc ABC, fig. 10, equals 9 1/2 inches, and the radii DA, DC equal each 7 inches required the area

9.5 X 3.5 = 33.25 inches the area.

Fig. 10.

To Find The Area Of A Segment Of A Circle

Rule. - Find the area of a sector whose are is equal to that of the given segment, and if it be less then a semicircle subtract the area of the triangle formed by the chord of the segment and radii of its extremities; but if more than a semicircle add the area of the triangle to the area of the sector and the remainder or sum is the area of the segment.

Thus suppose the area of the segment ABC E fig. 10. is required and that the length of the arc ABC equals 19 1/2 ft., DA and DC each equal 14 ft, and the chord AC equal 16 ft., 8 inches; also the perpendicular ED equal 7 1/2 feet.

16.666 X7 5

19.5 X 7 = 136.5 feet the area of the sector, ------------------=

2 62.49 feet the area of the triangl , 136.5 - 62.49 - 74 01 feet the area of the segment.

To Find The Area Of The Space Contained Between Two Concentric Circles Or The Area Of A Circular King

Rule. - 1 Multiply the sum of the inside and out side diameters by their difference and by .7854 the product is the area.

Rule 2. - The difference of the areas of the two circles will be the area of the Ring on space.

Suppose the external circle AD, fig. 11, equal 4 ft., and the internal circle BC 2 1/2 ft., required the area of the space contained between them or area of a Ring.

4 + 2.5 = 6.5 and 4 - 2.5 = 1.5 hence, 6.5 X 1.8 X .7854 =

7.65 ft., the area; or,

The area of 4 ft., is 12.566; the area of 2.5 is 4.9081. (See table of areas of circles.)

12.566 - 4.9081 = 7.6579 the area.

Fig. 11.

To Find The Area Of Ellipse Or Oval

Rule. - Multiply the diameter together and their product by .7854.

An oval is 20 inches by 15 inches what are its superficial contents 20 X 15 X .7854 = 235.62 inches the area.

To Find The Circumference Of An Ellipse Or Oval

Rule. - Multiply half the sum of the two diameters by 3.1416, the product will be the circumference.

Example. - An oval is 20 inches by 15 inches what is its circumference.

20 + 15

------------=17.5 X 3.1416 = 54.978 inches the circumference.