Circular segment: Difference between revisions

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{{short description|SliceArea ofbounded by a circlecircular cutarc perpendicularand toa thestraight radiusline}}

In [[geometry]], a '''circular segment''' (symbol: <span style="font-size:1.5em">⌓</span>) is a region of a [[circle]] which is "cut off" from the rest of the circle by a [[secant line|secant]] or a [[chord (geometry)|chord]]. More formally, a circular segment is a region of [[two-dimensional space]] that is bounded by an [[arc (geometry)|arc]] (of less than π radians by convention) of a circle and by the chord connecting the endpoints of the arc.

== Formulae ==

[[Image:Circularsegment.svg|frame|right|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).]]

In [[geometry]], a '''circular segment''' or '''disk segment''' (symbol: <span style="font-size:1.5em">⌓</span>) is a region of a [[disk (mathematics)|disk]]<ref>Mathematics distinguishes when necessary between the words ''circle'' and ''disk'': a disk is a plane area having a circle as its boundary, while a circle is the closed curve forming the boundary itself.</ref> which is "cut off" from the rest of the disk by a straight line. The complete line is known as a ''[[secant line|secant]]'', and the section inside the disk as a ''[[chord (geometry)|chord]]''.<ref>These terms refer to a line which intersects a curve. In this case, the curve is the circle forming the disk's boundary.</ref>

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 [[radian]]s, ''c'' the [[chord length]], ''s'' the [[arc length]], ''h'' the [[Sagitta (geometry)|sagitta]] ([[Height#In mathematics|height]]) of the segment, and ''a'' the [[area]] of the segment.

More formally, a circular segment is a [[Plane (mathematics)|plane region]] bounded by a [[circular arc]] (of less than π radians by convention) and the [[circular chord]] connecting its endpoints.

== Formulae ==

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 [[radian]]s, ''c'' the [[chord length]], ''s'' the [[arc length]], ''h'' the [[Sagitta (geometry)|sagitta]] ([[Height#In mathematics|height]]) of the segment, ''d'' the [[apothem]] 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 angle ===

The radius is:

:<math>R = \tfrac{h}{2}+\tfrac{c^2}{8h}</math><ref>The fundamental relationship between <math>R</math>, <math>c</math>, and <math>h</math> derivable directly from the Pythagorean theorem among <math>R</math>, C<math>c/2</math>, and r<math>R-h</math> as components of a right-angled triangle is: <math>R^2=(\tfrac{c}{2})^2+(R-h)^2</math> which may be solved for <math>R</math>, <math>c</math>, or <math>h</math> as required.</ref>

The central angle is

:<math> \theta = 2\arcsin\tfrac{c}{2R}</math>

=== Chord length and height ===

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The chord length is

:<math>c = 2R\sin\tfrac{\theta}{2} = R\sqrt{2(1-\cos\theta)}</math>

:<math>c = 2\sqrt{R^2 - (R - h)^2} = 2\sqrt{2Rh - h^2}</math>

The sagitta is

The [[Sagitta_(geometry)|sagitta]] is

:<math>h =R-\sqrt{R^2-\frac{c^2}{4}}= R(1-\cos\tfrac{\theta}{2})=R\left(1-\sqrt{\tfrac{1+\cos\theta}{2}}\right)=\frac{c}{2}\tan\frac{\theta}{4}</math>

The [[apothem]] is

:<math> d = R - h = \sqrt{R^2-\frac{c^2}{4}} = R\cos\tfrac{\theta}{2} </math>

=== Arc length and area ===

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:<math>s = {\theta}R</math>

The area ''<math>a''</math> of the circular segment is equal to the area of the [[circular sector]] minus the area of the triangular portion (using the double angle formula to get an equation in terms of <math>\theta</math>):

:<math>a = \tfrac{R^2}{2} \left(\theta - \sin \theta\right)</math>

In terms of {{math|''c''}} and {{math|''R''}},

:<math>a = \tfrac{R^2}{2} \left(2\arcsin\tfrac{c}{2R} - \sin\left(2\arcsin\tfrac{c}{2R}\right)\right) = R^2\left(\arcsin\frac{c}{2R} - \frac{c}{2R}\sqrt{1-\left(\frac{c}{2R}\right)^2}\right)</math>

In terms of {{math|''R''}} and {{math|''h''}},

:<math>a = R^2\arccos\left(1-\frac{h}{R}\right) - \left(R-h\right)\sqrt{R^2-\left(R-h\right)^2}</math>

In terms of {{math|''c''}} and {{math|''h''}},

Unfortunately, <math>a</math> is a [[transcendental function]] of <math>c</math> and <math>h</math> 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 asymptotically approaches <math>\tfrac{2}{3}c\cdot h</math>. If <math>\theta<<1, a=\tfrac{2}{3}c\cdot h</math> is a substantially good approximation.

:<math>a = \left(\frac{c^2+4h^2}{8h}\right)^2\arccos\left(\frac{c^2-4h^2}{c^2+4h^2}\right) - \frac{c}{16h}(c^2-4h^2)</math>

As the central angle approaches π, the area of the segment is converging to the area of a semicircle, <math>\tfrac{\pi R^2}{2}</math>, so a good approximation is a delta offset from the latter area:

Unfortunately, <math>a</math> is a [[transcendental function]] of <math>c</math> and <math>h</math> so no algebraic formula in terms of these can be stated. But whatWhat can be stated is that as the central angle gets smaller (or alternately the radius gets larger) , the area ''a'' rapidly and asymptotically approaches <math>\tfrac{2}{3}c\cdot h</math>. If <math>\theta \ll 1<<1/math>, <math>a = \tfrac{2}{3}c\cdot h</math> is a substantially good approximation.

:<math>a\approx \tfrac{\pi R^2}{2}-(R+\tfrac{c}{2})(R-h)</math> for h>.75''R''

If <math>c</math> is held constant, and the radius is allowed to vary, then we have<math display="block">\frac{\partial a}{\partial s} = R</math>

=== Etc. ===

The perimeter ''p'' is the arclength plus the chord length,

As the central angle approaches π, the area of the segment is converging to the area of a [[semicircle]], <math>\tfrac{\pi R^2}{2}</math>, so a good approximation is a delta offset from the latter area:

:<math>p=c+s=c+\theta R</math>

:<math>a\approx \tfrac{\pi R^2}{2}-(R+\tfrac{c}{2})(R-h)</math> for h>.75''R''

As a proportion of the whole area of the disc, <math>A= \pi R^2</math>, you have

As an example, the area is one quarter the circle when ''θ'' ~ 2.31 radians (132.3°) corresponding to a height of ~59.6% and a chord length of ~183% of the radius.{{Clarify|date=December 2021|reason= A diagram with these numbers would be a good addition to the example}}

=== Other properties ===

The perimeter ''p'' is the arclength plus the chord length,:

:<math>p=c+s=c+\theta R</math>

Proportion of the whole area of the circle:

:<math> \frac{a}{A}= \frac{\theta - \sin \theta}{2\pi}</math>

== Applications ==

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

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

For calculating the area or locating the centroid of a planar shape that contains circular segments.

== See also ==