Length — Question 9

Question 9

Accurate chords can still underestimate a curved length

A circular boundary has radius $R=1.000\ \mathrm{m}$. A survey approximates its circumference using $N$ equal straight chords joining equally spaced points. Take the chord-sum model and exact circumference to be $$L_N=2NR\sin(180^\circ/N),\qquad C=2\pi R.$$ You may use $\sin30^\circ=0.5$, $\sin15^\circ\approx0.258819$, and $\sin7.5^\circ\approx0.130526$. These approximations are sufficient for the requested comparisons; no trigonometric derivation is required.

1. Compute $L_N$ for $N=6,12,24$ and the percentage shortfall $100(C-L_N)/C$. Which of these choices meets a requirement of less than $1\%$ geometric error?

2. Explain why even perfectly measured chords underestimate this boundary’s length. Would a finer ruler alone correct the discrepancy at fixed $N$?

3. A different, unknown boundary is sampled at successively finer spacing, giving increasing chord sums. Does that observation alone prove the length is infinite, or establish that the latest sum is within $1\%$ of the true length? Explain the missing evidence.

Question 9 - Solution

Accurate chords can still underestimate a curved length

(a) Compare the same target length at each sampling density.

$$L_N=2NR\sin(\pi/N),\qquad\varepsilon_N=100\left(1-\frac{L_N}{2\pi R}\right)\%.$$

The circumference is $C=2\pi\ \mathrm{m}\approx6.283185\ \mathrm{m}$. The supplied sine values give: $$\begin{aligned}L_6&=6.000\ \mathrm m,&\varepsilon_6&\approx4.51\%,\\L_{12}&=6.211656\ \mathrm m,&\varepsilon_{12}&\approx1.14\%,\\L_{24}&=6.265248\ \mathrm m,&\varepsilon_{24}&\approx0.285\%.\end{aligned}$$ Of the listed choices, 24 chords meet the less-than-$1\%$ requirement. Rounding the sine values as supplied does not alter this decision.

(b) Improve the geometric approximation, not just the readout.

$$\sin z<z\ (0<z<\pi/2)\ \Longrightarrow\ L_N<2NR\frac{\pi}{N}=2\pi R=C.$$

A straight chord is shorter than the circular arc connecting its endpoints. Adding precisely measured chords therefore still omits curved distance. A finer ruler can reduce chord-measurement error but does not replace chords by arcs. Increasing $N$, or applying a justified model-based correction, addresses this separate geometric discrepancy.

(c) More samples do not supply a universal convergence guarantee.

The circle itself is a counterexample to the claim of infinity: its chord sums increase yet approach the finite value $2\pi R$. For an unknown boundary, unresolved bends may add further length. A few increasing sums do not bound that missing contribution.

A defensible accuracy claim needs additional information, such as a validated shape or curvature bound, an appropriate convergence/error model, and a clear definition of the spatial scale at which the boundary is to be measured. The latest chord sum alone establishes neither infinite length nor a $1\%$ error limit.