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

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 (or apothem) 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:

${\displaystyle (2R-h)\cdot h={\frac {c}{2}}\cdot {\frac {c}{2}}={\frac {c^{2}}{4}}}$

${\displaystyle 2R={\frac {c^{2}}{4h}}+h}$

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

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

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

Area

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

${\displaystyle A={\frac {R^{2}}{2}}\left(\theta -\sin \theta \right)=R^{2}\left(\arcsin {\frac {c}{2R}}-{\frac {c}{2R}}{\sqrt {1-\left({\frac {c}{2R}}\right)^{2}}}\right)=R\left(R\arccos {\frac {d}{R}}-d{\sqrt {1-{\frac {d^{2}}{R^{2}}}}}\right)}$

with the central angle in radians, or

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

with the central angle in degrees.

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

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

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.

One can reconstruct the full dimensions of a complete circular object from fragments by measuring the arc length and the chord length of the fragment.

To check hole positions on a circular pattern. Especially useful for quality checking on machined products.

For calculating the area or centroid of a polygon that contains circular segments.

See also

• Chord (geometry)
• Spherical segment
• Conic section
• Cross section

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