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Circular segment

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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.

Contents

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 = h 2 + c 2 8 h

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 = α 180 π R = θ R = arcsin ( c h + c 2 4 h ) ( h + c 2 4 h )

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 some of the trigonometric forms below.

The chord length is

c = 2 R sin θ 2 = R 2 2 cos θ = 2 R 1 ( d / R ) 2

The sagitta is

h = R ( 1 cos θ 2 ) = R R 2 c 2 4

The angle is

θ = 2 arctan c 2 d = 2 arccos d R = 2 arccos ( 1 h R ) = 2 arcsin c 2 R

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 = R 2 2 ( θ sin θ )

with the central angle in radians, or

A = R 2 2 ( α π 180 sin α )

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

Circular segment Wikipedia


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