-- Approximating the area of a circular segment
David W. Cantrell
a circle can be expressed precisely in terms of elementary functions:
A = ( (c^2 + 4h^2)^2 arctan(2h/c)/(2h) - c(c^2 - 4h^2) )/(16h) [0]
[For a figure, see
<http://mathforum.org/dr.math/faq/formulas/faq.circle.html#segment>.]
Quadratic approximations for the area of minor segments were investigated
by Russell A. Gordon in "Squaring a Circular Segment", College Math. J.
39:3 (May 2008) pp. 212-220. In this article, two simple rational
approximations are given. The first, for minor segments, has much greater
accuracy than the approximations mentioned by Gordon; the second works
reasonably well for all segments, minor or major (i.e., resp., <= or >= a
semicircle, so to speak).
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A little more background
Of the quadratic approximations obtained by Gordon, my favorite, for
aesthetic reasons, is
2/3 c h + (pi/2 - 4/3) h^2. [1]
It gives the area precisely at the extremes for a minor segment:
when c = 2h, in which case the segment is a semicircle; and
when h = 0, in which case the segment is degenerate, having area 0.
(It might be noted that A -> 0 as h -> 0 is not immediately obvious from
formula [0].) Gordon also shows nice geometric interpretations of [1].
But unfortunately, at its worst, the error of [1] is about 3.9%.
The last two quadratic approximations obtained by Gordon are more accurate,
having worst error of about 1.9%.
Gordon also mentions one rational approximation
2/3 c h + 8/15 h^3/c
having again worst error of about 1.9%. But by changing the second
coefficient, giving
2/3 c h + 1/2 h^3/c,
an approximation mentioned on p. 92 of _Handbook of Mathematics and
Computational Science_, J. Harris and H. Stocker (Springer, 1998),
the worst error then becomes about 0.8%.
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A new rational approximation for the area of a minor segment:
2/3 c h + h^3/( 1.75835(c - 2h) + 6h/(3pi - 8) ) [2]
This approximation has |relative error| < 0.035%. It has the nice feature,
like Gordon's [1], that area is given precisely at both extremes for a
minor segment.
The coefficient 1.75835 was determined numerically; those who would prefer
to use a rational number may use 51/29 instead.
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A new rational approximation for the area of any segment:
pi h ( h/4 + 2c^2/(3pi(c - 2h) + 32h) ) [3]
Of course it is substantially more difficult to approximate well the area
of general segments, major or minor, than to approximate the area of just
minor segments. Our approximation [3] has |relative error| < 1.4%. It gives
the area precisely in the two degenerate cases, h = 0 and c = 0, as well as
in the semicircular case, c = 2h.
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By the way, if we need to approximate the area of major segments better
than [3] allows, then [2] can be used "indirectly":
The radius of the circle can be calculated using
r = (c^2 + 4h^2)/(8h)
and then we can calculate the area of the full circle. The height h' of the
corresponding minor segment, on the other side of the chord, is
h' = 2r - h = c^2/(4h)
Now use h' instead of h in [2], approximating the area of the minor
segment. Finally, subtract that approximation from the area of the full
circle, thereby obtaining an approximation for the area of the major
segment, with |relative error| < 0.035%.
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Comments are welcomed.
David W. Cantrell
David W. Cantrell
Another simple approximation for the area of a general circular segment is
given. This new approximation has |relative error| < 0.46%.
Also, a silly mistake at the end of my original post is corrected.
I should not have just parroted the error bound for area of a minor
segment! Using the procedure above, which approximates the area of a major
segment, we do better: |relative error| < 0.023%.
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A new algebraic approximation for the area of any segment:
h sqrt( (2/3 c)^2 + (3 pi^2/32 - 8/9) c h + (pi/4 h)^2 ) [4a]
or written alternatively,
h sqrt( 4/9 c (c - 2h) + pi^2/16 h (3/2 c + h) ) [4b]
This approximation has |relative error| < 0.46%. It, like the rational
approximation [3], gives the area precisely in the two degenerate cases,
h = 0 and c = 0, as well as in the semicircular case, c = 2h.
David W. Cantrell