Question

What is a segment in a circle.

Answer 1

Any portion of the circle is segment.


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Answer 2

In geometry, a circular segment (symbol: ⌓) is a region of a circle which is "cut off" from the rest of the circle by a secant or a chord. More formally, a circular segment is a region of two-dimensional space that is bounded by an arc (of less than 180°) of a circle and by the chord connecting the endpoints of the arc.

FormulaEdit



A circular segment (in green) is enclosed between a secant/chord (the dashed line) and the arc whose endpoints equal the chord's (the arc shown above the green area).

Let R be the radius of the circle, θ the central angle in radians, α is the central angle in degrees, c the chord length, s the arc length, h the sagitta (height) of the segment, and d the height of the triangular portion.

The radius is

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

The radius in terms of h and c can be derived above by using the Intersecting Chords Theorem, where 2R (the diameter) and c are perpendicularly intersecting chords:

{\displaystyle (2R-h)*h={\frac {c}{2}}*{\frac {c}{2}}={\frac {c^{2}}{4}}}{\displaystyle 2R={\frac {c^{2}}{4h}}+h}{\displaystyle R={\frac {c^{2}}{8h}}+{\frac {h}{2}}}

The arc length is

{\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 be derived above by considering an inscribed angle that subtends the same arc, and one side of the angle is a diameter. The angle thus inscribed is θ/2 and is part of a right triangle whose hypotenuse is the diameter. This is also useful in deriving other inverse trigonometric forms below.

With further aid of half-angle formulae and pythagorean identities, the chord length is

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

The sagitta is

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

The angle is

{\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}}}

AreaEdit

The area A of the circular segment is equal to the area of the circular sectorminus the area of the triangular portion

{\displaystyle A={\frac {R^{2}}{2}}\left(\theta -\sin \theta \right)}

with the central angle in radians, or

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

with the central angle in degrees.

As a proportion of the whole area of the disc, {\displaystyle S=\pi R^{2}}, you have

{\displaystyle {\frac {A}{S}}={\frac {1}{2\pi }}\left(\theta -\sin \theta \right)={\frac {\alpha }{360^{\circ }}}-{\frac {\sin \alpha }{2\pi }}}







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