Suppose you could have an arc a, a portion of a circle of radius r, and you understand two issues: the size c of the chord of the arc, and the size b of the chord of half the arc, illustrated beneath.
Right here θ is the central angle of the arc. Then the size of the arc, rθ, is roughly
a = rθ ≈ 12 b²/(c + 4b).
If the arc is reasonably small, the approximation may be very correct.
This approximation is easy, correct, and never apparent, very like the one in this publish
Derivation
Let φ = θ/4. Then the angle between the chords b and c is φ. This follows from the inscribed angle theorem, illustrated beneath.
There are two proper triangles within the diagram above which have an angle φ: a smaller triangle with hypotenuse b and a bigger triangle with hypotenuse 2r. From the smaller triangle we be taught
cos(φ) = c / 2b
and from the bigger triangle we be taught
sin(φ) = b / 2r.
Now broaden in energy collection.
c / 2b = cos(φ) = 1 − φ2/2! + φ4/4! − …
2b / a = sin(φ) / φ = 1 − φ2/3! + φ4/5! − …
If we multiply 2b / a by 3 and subtract c / 2b then the φ2 phrases cancel out and we get
6b / a − c / 2b = 2 − φ4/60 + …
and so
6b / a − c / 2b ≈ 2
to a really excessive diploma of accuracy when φ is small. The approximation follows by fixing for a.
Instance
Let θ = π/3 and so φ = 0.26…, not a very small worth of φ, however sufficiently small for the approximation to work properly.
Set r = 1 so a = θ. Then
b = 2 sin(π/12) = 0.51764
and
c = 2b cos(π/12) = 1.
Now in utility, we all know b and c, not θ, and so fake we measured b = 0.51764 and c = 1. Then we might approximate a by
12b²/(c + 4b) = 1.04718
whereas the precise worth is 1.04720. Except you may measure lengths to greater than 4 vital figures, the approximation could has properly be precise as a result of approximation error could be lower than measurement error.
[1] J. M. Bruce. Approximation to a Round Arc. The American Mathematical Month-to-month. Vol. 49, No. 3 (March 1942), pp. 184–185
