Formula Circular segment

let r radius of circle, θ central angle in radians, α central angle in degrees, c chord length, s arc length, h sagitta (height) of segment, , d height of triangular portion.

the radius is

r
=
h
+
d
=


h
2


+



c

2



8
h





{\displaystyle r=h+d={\frac {h}{2}}+{\frac {c^{2}}{8h}}}

the radius in terms of h , c can derived above using intersecting chords theorem, 2r (the diameter) , c perpendicularly intersecting chords.

the arc length is

s
=


α

180

∘




π
r
=

θ

r
=
arcsin
⁡

(


c

h
+



c

2



4
h






)


(
h
+



c

2



4
h



)



{\displaystyle s={\frac {\alpha }{180^{\circ }}}\pi r={\theta }r=\arcsin \left({\frac {c}{h+{\frac {c^{2}}{4h}}}}\right)\left(h+{\frac {c^{2}}{4h}}\right)}

the arc length in terms of arcsin can derived above considering inscribed angle subtends same arc, , 1 side of angle diameter. angle inscribed θ/2 , part of right triangle hypotenuse diameter. useful in deriving other inverse trigonometric forms below.

with further aid of half-angle formulae , pythagorean identities, chord length is

c
=
2
r
sin
⁡


θ
2


=
r


2
−
2
cos
⁡
θ


=
2
r


1
−
(
d

/

r

)

2






{\displaystyle c=2r\sin {\frac {\theta }{2}}=r{\sqrt {2-2\cos \theta }}=2r{\sqrt {1-(d/r)^{2}}}}

the sagitta is

h
=
r

(
1
−
cos
⁡


θ
2


)

=
r
−



r

2


−



c

2


4






{\displaystyle h=r\left(1-\cos {\frac {\theta }{2}}\right)=r-{\sqrt {r^{2}-{\frac {c^{2}}{4}}}}}

the angle is

θ
=
2
arctan
⁡


c

2
d



=
2
arccos
⁡


d
r


=
2
arccos
⁡

(
1
−


h
r


)

=
2
arcsin
⁡


c

2
r





{\displaystyle \theta =2\arctan {\frac {c}{2d}}=2\arccos {\frac {d}{r}}=2\arccos \left(1-{\frac {h}{r}}\right)=2\arcsin {\frac {c}{2r}}}



area

the area of circular segment equal area of circular sector minus area of triangular portion

a
=



r

2


2



(
θ
−
sin
⁡
θ
)



{\displaystyle a={\frac {r^{2}}{2}}\left(\theta -\sin \theta \right)}

with central angle in radians, or

a
=



r

2


2



(



α
π


180

∘




−
sin
⁡
α
)



{\displaystyle a={\frac {r^{2}}{2}}\left({\frac {\alpha \pi }{180^{\circ }}}-\sin \alpha \right)}

with central angle in degrees.

applications

the area formula can used in calculating volume of partially-filled cylindrical tank.

in design of windows or doors rounded tops, c , h may known values , can used calculate r draftsman s compass setting.