Circular arc approximation

Suppose you have an arc a, a portion of a circle of radius r, and you know two things: the length c of the chord of the arc, and the length b of the chord of half the arc, illustrated below. Here θ is the central angle of the arc. Then the length of the arc, rθ, is approximately a = rθ ≈ b²/(c + 4b). If the arc is moderately small, the approximation is very accurate. This approximation is simple, accurate, and not obvious, much like the one in this post Derivation Let φ = 4θ. Then the angle between the chords b and c is φ. This follows from the inscribed angle theorem, illustrated below. There are two right triangles in the diagram above that have an angle φ: a smaller triangle with hypotenuse b and a larger triangle with hypotenuse 2r. From the smaller triangle we learn cos(φ) = c / 2b and from the larger triangle we learn sin(φ) = b / 2r. Now expand in power series. c / 2b = cos(φ) = 1 − φ2/2! + φ4/4! − … 2b / a = sin(φ) / φ = 1 − φ2/3! + φ4/5! − … If we multiply 2b / a and subtract…