Machinery's Handbook, 31st Edition
SEGMENTS OF CIRCLES 83 Segments of a Circle.— The table that follows gives the principal formulas for dimen- sions of circle segments. The dimensions are illustrated in the figures on pages 79 and 84 . When two of the dimensions found together in the first column are known, the other dimensions are found by using the formulas in the corresponding row. For ex- ample, if radius r and chord c are known, solve for angle a using Equation (13), then use Equations (14) and (15) to solve for h and l , respectively. In these formulas, the value of a is in degrees between 0 and 180 ° . Formulas for Segments of a Circle Given Formulas a , r (1) (2) (3) c 2 r l
α 2 --
α 2 = sin --
π r α 180 = -----
h r 1– cos =
c
π c α 360sin = ----------- α 2 -- -----
-- -----
2sin = -------- α 2
r
l
α 4 --
(4)
(5)
(6)
2 = –-- tan 2 h tan = ------
h c
a , c
π h α α 2 – cos -- = ------------------- 180 1 ----- 180 l 1 α 2 – cos -- πα = -------------------- -----
h 1– cos = ----------- α 2 -- ----
α 4 -- ----
c
l
r
(7)
(8)
(9)
a , h
α 2 -- ---------
360 l sin πα = ----------
180 π -----
l α --
(10)
(11)
(12)
a , l
=
r
c
h
cos –1 1 c 2 2 r 2 – ----
π 90 --- r sin –1
(13)
(14)
(15)
c 2 r ---
2
2
r, c
–
4 r
c
=
=
α
l
2 = – ------------
h r
π 90 --- r cos –1 1
(16)
(17)
(18)
h r – --
h r – --
r, h
α
2cos –1 1
2 h 2 r h – ( ) =
=
c
=
l
90 l π R = sin ----
90 l π r – cos ----
180 π -----
l r -
(19)
(20)
(21)
h r 1 =
c 2 r
r, l
α
=
4tan –1 2 h c ---
2
2
(22)
(23)
(24)
c 2 4 h 2 + 360 h ----------
2 h c ---
=
tan –1
c, h
c + 8 h = ---------- 4 h
α
π
=
l
r
Given
Formula To Find
Given
Formula To Find
(26) Solve Equation (26) for a by iteration a , then α 1– cos = ----------- α 2 -- ---- 180 π ----- l h --
360 π ----- l c -
α = ------
----
(25) Solve Equation (25) for a by iteration a , then α 2 sin --
h, l
c, l
r = Equation (10) c = Equation (11)
r = Equation (10) h = Equation (5)
a Equations (25) and (26) cannot be easily solved by ordinary means. To solve these equations, test various values of α until the left side of the equation equals the right side. For example, if given c = 4 and l = 5, the left side of Equation (25) equals 143.24, and testing various values of α reveals the right side equals 143.24 when α = 129.62°. Angle a is in degrees, 0 < a < 180 Formulas for Circular Segments contributed by Manfred Brueckner.
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