TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-13

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The arc length of a unit semicircle is pi

Statement

Every once-traversed semicircle of radius $1$ has arc length $π$ . In particular, the upper unit semicircle $γ (t) = (\cos t, \sin t)$ , $0 \leq t \leq π$ , has length $π$ .

Facts & Assumptions

Given: The path $γ (t) = (\cos t, \sin t)$ on $[0, π]$ .

[L1]

Vector differentiation is componentwise (The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral).

[L2]

The functions sine and cosine are differentiable, with $(\sin t)^{'} = \cos t$ and $(\cos t)^{'} = - \sin t$ (The derivatives of sine and cosine are cosine and minus sine).

[L3]

For every real $t$ , $\sin^{2} t + \cos^{2} t = 1$ (Parity and the Pythagorean identity for sine and cosine).

[L4]

A $C^{1}$ path has length equal to the integral of its Euclidean speed (If $γ : [a, b] \to R^{n}$ is continuous, differentiable on $(a, b)$ , and $γ^{'}$ extends continuously to $[a, b]$ , then $L (γ) = \int_{a}^{b} ∥ γ^{'} (t) ∥_{2} d t$ ).

[L5]

The integral of the constant function $1$ on $[a, b]$ is $b - a$ (If $m \leq f \leq M$ on $[a, b]$ then $m (b - a) \leq L (f, P) \leq \underline{\int_{a}^{b}} f \leq \overline{\int_{a}^{b}} f \leq U (f, P) \leq M (b - a)$ for every partition $P$ ; in particular every constant function is integrable, with $\int_{a}^{b} c = c (b - a)$ ).

[L6]

Path length is invariant under every continuous surjective monotone reparametrization (Arc length is invariant under every continuous surjective monotone reparametrization, including pauses and reversal).

Proof

technique · direct

1.1

By [L1] and [L2], $γ^{'} (t) = (- \sin t, \cos t)$ on $[0, π]$ .

givenL1L2

2.1

By [L3], $∥ γ^{'} (t) ∥_{2} = \sqrt{\sin^{2} t + \cos^{2} t} = 1$ .

step 1.1L3algebra

3.1

By [L4] and [L5], $L (γ) = \int_{0}^{π} 1 d t = π$ .

step 2.1L4L5

4.1

Translating or rotating the displayed path does not change the differences between its points, and reversing or monotonically reparametrizing it does not change its length by [L6]. Thus every once-traversed unit semicircle has length $π$ .

step 3.1L6algebra∎

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Sources

J. Lebl, Basic Analysis II, section 11.4.3 (standard reference, not scraped)