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 Π radians by convention) of a circle and by the chord connecting the endpoints of the arc.

FormulaeEdit

 
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 arc which forms part of the perimeter of the segment, θ the central angle subtending the arc in radians, c the chord length, s the arc length, h the sagitta (height) of the segment, and a the area of the segment.

Usually, chord length and height are given or measured, and sometimes the arc length as part of the perimeter, and the unknowns are area and sometimes arc length. These can't be calculated simply from chord length and height, so two intermediate quantities, the radius and central angle are usually calculated first.

Radius and central angleEdit

The radius is:[1]

 

The central angle is

 

To convert angles to degrees, use   in place of   in the formulae.

Chord length and heightEdit

The chord length and height can be back-computed from radius and central angle by:

The chord length is

 

The sagitta is

 

Arc length and areaEdit

The arc length, from the familiar geometry of a circle, is

 

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

 

Unfortunately,   is a transcendental function of   and   so no algebraic formula in terms of these can be stated. But what can be stated is that as the central angle gets smaller (or alternately the radius gets larger) , the area a rapidly and asymtotically approaches  . If   is a substantially good approximation.

Etc.Edit

As a proportion of the whole area of the disc,  , you have

 

ApplicationsEdit

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

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 planar shape that contains circular segments.

See alsoEdit

ReferencesEdit

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

External linksEdit

  1. ^ 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: