arc \(s\), chord \(a\), sagitta \(c\), diameter \(d\)
Geometry
Signed relative error of \(2c^{2}/d + a\)
Signed relative error (%) as a function of the central angle \(2\theta\). It is independent of \(d\). Hover for values; the marker tracks the slider.
| central angle 2θ | s (d=1) | approximation | signed relative error |
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Let \(d\) be the diameter (radius \(r=d/2\)) and \(2\theta\) the central angle (\(0<\theta\le\pi/2\)). The arc, chord, and sagitta are
$$ s = d\,\theta, \qquad a = d\sin\theta, \qquad c = \frac{d}{2}\bigl(1-\cos\theta\bigr). $$Taylor expansion around \(\theta=0\) gives
$$ a = d\Bigl(\theta - \frac{\theta^{3}}{6} + \frac{\theta^{5}}{120} - \cdots\Bigr), \qquad 1-\cos\theta = \frac{\theta^{2}}{2} - \frac{\theta^{4}}{24} + \cdots $$so the correction term is
$$ \frac{2c^{2}}{d} = \frac{d}{2}\bigl(1-\cos\theta\bigr)^{2} = \frac{d\,\theta^{4}}{8} - \frac{d\,\theta^{6}}{48} + \cdots $$and the arc-length approximation formula becomes
$$ \frac{2c^{2}}{d} + a = d\Bigl(\theta - \frac{\theta^{3}}{6} + \frac{\theta^{4}}{8} + \frac{\theta^{5}}{120} - \cdots\Bigr). $$The signed relative error against the true arc length \(s=d\theta\) is
$$ \frac{\text{error}}{s} = \frac{2c^{2}/d + a - s}{s} = -\frac{\theta^{2}}{6} + \frac{\theta^{3}}{8} + O(\theta^{4}). $$The arc-length approximation appears in Dream Pool Essays/Mengxi Bitan (『夢渓筆談』, c. 1088) of Shen Kuo (沈括, 1031–1095). Guo Shoujing (郭守敬) and their collaborators used the method for the Shoushi calendar (授時曆), which was adopted in 1281 as the official Chinese calendar.
Other Sources:
J.-P. J. Chen, "The evolution of transformation media in spherical trigonometry in 17th- and 18th-century China, and its relation to 'Western learning'", Historia Mathematica 37 (2010) 62–109;
D. B. Wagner, "Shen Gua and an ignorant editor on the length of an arc", research note (2012);
H. Shinkai, "Deciphering Seki Takakazu's Juji Hatsumei (Shoushi Hatsumei)" (2026).