Machinery's Handbook, 31st Edition
50
CIRCLE FORMULAS
π L R 180
= ·
Central angle , in degrees φ
π 180
Arc length L R φ = ·
2
2
L φ
K
E F F + 4 8
π 180
2
= ·
=
=
Radius R
L
RL
π
2
=
·
=
Area of sector K R φ
360 2
RL E R F − 2 2 ( ) –
=
Area of segment S
2 φ ( ) ( ( )) sin c os φ 2
E F R F D = − = 2 2 ( )
Chord length
2
2
R E −
4
F R
R = − 1
= −
Depth
2
φ 2 ( ) ( ) φ
/ E R F E R 2 −
Fig. 6b.
=
tan
=
sin
2 2
Annulus R 1 = radius of outer circle R 2 = radius of inner circle Area of annulus = π
(
)
2
2
W R R −
1
2
2 − ( )
φ
π R R 1 2 2
U
=
Area of annulus segment
360
Fig. 6c. Example 2: Find the area of a sector of a circle having a central angle of 30 ° and a radius of 7 cm. Solution: Referring to Fig. 6b, K φ° 360 ----- π R 2 30 360 ----- π 7 2 12.83 cm 2 = = = × × Example 3: Find the chord length E of a circular segment (Fig. 6b) with a depth of 2 cm at the center that is formed in a circle whose radius 12 cm. Solution: The chord length is E 2 F 2 R F – ( ) 2 2 44 = = = = 4 11 = 13.27 cm 2[(2)(12) – 2] Example 4: Find the area S of the segment in Example 3. Solution: First determine angle f , then find arc length L of the segment, and then solve for area S, as follows:
E 2 ⁄ R F – ------- 13.27 2 ⁄ 12– 2 =
φ 2 -- tan
φ 2 --
---------- 0.6635, =
=
=
33.56 °,
67.13 ° =
φ
tan –1 0.6635 =
π
π
L R φ = × = 12 × 67.13° ×
= 14.06 cm
180
180
Area S RL 2 = – ----------- 12 14.06 ( ) 2 = Copyright 2020, Industrial Press, Inc. ---- E R F – ( ) 2
– ------------ 84.36 – 66.35 18.01 cm 2 = = ebooks.industrialpress.com
------------ 13.27 10 ( ) 2
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