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Theorem 1: Length of Arc


Let P be the point P(1, 0).

For every number s between 0 and 2π there is a point Q on the unit circle such that the arc PQ has length s.


We give the proof for s between 0 and π/2, whence

0 ≤ h ≤ π/4.

Let A(x) be the area of the sector POQ where Q = Q(x, y) (Figure 7.1.4). Then A(0) = π/4, A(1) = 0 and the function A(x) is continuous for 0 ≤ x ≤ 1. By the Intermediate Value Theorem there is a point x0 between 0 and 1 where the sector has area ½s

A(x0) = ½s.

Therefore the arc PQ has length

2A(x0) = s.


x = 0, x = x0, x = 1

Figure 7.1.4

Arc lengths are used to measure angles. Two units of measurement for angles are radians (best for mathematics) and degrees (used in everyday life).

Last Update: 2006-11-05