Circular segment: Difference between revisions

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== 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.

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=== 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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:<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''}},

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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>

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

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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}}

=== Etc.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:

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

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

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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 ==