Circular segment

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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[edit]

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 is 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 

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

  • The arc length is 

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 
  • The sagitta is 
  • The angle is 

Area[edit]

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

with the central angle in radians, or

with the central angle in degrees.

Applications[edit]

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 the draftsman's compass setting.

See also[edit]

References[edit]

External links[edit]