Circular segment

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.

Formula

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

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.

The arc length is

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. This is also useful in deriving some of the trigonometric forms below.

The chord length is

c = 2R\sin\frac{\theta}{2} = R\sqrt{2-2\cos\theta}

The sagitta is

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

The angle is

\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 A of the circular segment is equal to the area of the circular sector minus the area of the triangular portion—that is,

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

with the central angle in radians, or

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

with the central angle in degrees.

Applications

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

In the design of windows or doors with rounded tops, c and h may be the only known values and can be used to calculate R for the draftsman's compass setting.

References

Weisstein, Eric W. "Circular segment". MathWorld.

External links

Definition of a circular segment With interactive animation
Formulae for area of a circular segment With interactive animation

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