DIMENSIONS OF PLANE FIGURES Machinery's Handbook, 31st Edition
79
Segment of a Circle: l
α = angle, in degrees
A = area
l = length of arc
1 ⁄ 2 rl c r h – ( ) – [ ] =
2 h 2 r h – ( ) =
c
A
h
c 2 4 h 2 + 8 h = ----------
0.01745 r α =
r
l
α c
57.296 l r = ----------
r
1 ⁄
r 1 α 2 ⁄ ( ) – cos [ ]
α
2
c 2 –
h r =
–
4 r
=
2
See also Segments of a Circle starting on page 83. Example: The radius r is 60 inches and the height h is 8 inches. Find the length of the chord c . c 2 h 2 r h – ( ) 2 8 2 60– 8 ( × ) × 2 896 2 29.93 × 59.86 in. = = = = = Example: If c = 16, and h = 6 inches, what is the radius of the circle of which the segment is a part? r c 2 4 h 2 + 8 h ---------- 16 2 4 6 2 × + 8 6 × --------------- 256 + 144 48 ------------ 400 48 ----- 8 1 ⁄ 3 in. = = = = = Cycloid: l d r Area A 3 π r 2 9.4248 r 2 2.3562 d 2 = = = = 3 area of generating circle × = Length of cycloid l 8 r 4 d = = = See also Areas Enclosed by Cycloidal Curves on page 73. Example: The diameter of the generating circle of a cycloid is 6 inches. Find the length l of the cycloidal curve and the area enclosed between the curve and the base line. l 4 d 4 6 × 24 in. = = = A 2.3562 d 2 2.3562 6 2 × 84.82 in. 2 = = = Circular Ring (Annulus):
π R 2 r 2 – ( ) = 3.1416 R r + ( ) R r – ( ) = 0.7854 D 2 d 2 – ( ) =
3.1416 R 2 r 2 – ( )
Area
A = =
D d r
R
0.7854 D d + ( ) D d – ( ) =
Example: Let the outside diameter D = 12 centimeters and the inside diameter d = 8 centimeters. Find the area of the ring. A 0.7854 D 2 d 2 – ( ) 0.7854 12 2 8 2 – ( ) 0.7854 144 – 64 ( ) 0.7854 80 × = = = = 62.83cm 2 = By the alternative formula: A 0.7854 D d + ( ) D d – ( ) 0.7854 12 + 8 ( ) 12 – 8 ( ) 0.7854 20 × 4 × = = = 62.83cm 2 = Sector of Circular Ring :
A = area,
α = central angle, in degrees
απ 360 ----- R 2 r 2 – ( ) 0.00873 α R 2 r 2 – ( ) = = απ 4 360 × --------- D 2 d 2 – ( ) 0.00218 α D 2 d 2 – ( ) = =
A
R
r
d
D
Example: Find the area, if the outside radius R = 5 inches, the inside radius r = 2 inches, and a = 72 degrees. A 0.00873 α R 2 r 2 – ( ) 0.00873 72 5 2 2 2 – ( ) × = = 0.6286 25 – 4 ( ) 0.6286 21 × 13.2 in. 2 = = =
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