pi: the Equivalent Characterizations

Let $c=(c_1,c_2)\in {\mathbb{R}}^2$ and $r>0$. A circular arc of the circle with centre $c$ and radius $r$ is a restriction

${\gamma }_{c,r}{\mathrm{\mid }}_{[\alpha ,\beta ]},{\gamma }_{c,r}(t)=c+r(\cos t,\sin t),\alpha <\beta ,$

taken as a parametrized path rather than as its image; its trace is that image. The parameter interval is part of the arc, because a set of points does not determine a length: the same trace is swept by restrictions of different lengths.

The arc's length is the path length of Paths in ${\mathbb{R}}^n$, inscribed polygonal sums, arc length as their supremum, and rectifiability computed with the Euclidean norm of The $p$-norms $\parallel x{\parallel }_p$ for rational $p\ge 1$, and $\parallel x{\parallel }_{\mathrm{\infty }}$. The circumference $C(r)$ is $L({\gamma }_{c,r})$ for the once-around parameter interval $[0,2\pi ]$, where $\pi$ is the constant of Pi as twice the smallest positive zero of cosine. Translation does not affect the value, so the notation suppresses $c$. The diameter is $d(r):=2r$.

The phrase once around is part of the convention: a parametrized path that repeats the same trace can have a larger length.

Let $a<b$, and let $g,f:[a,b]\to \mathbb{R}$ be continuous with $g(x)\le f(x)$ for every $x$. The Riemann area between their graphs is

$\mathrm{area}\{(x,y):a\le x\le b,g(x)\le y\le f(x)\}:={\int }_a^b(f(x)-g(x))dx\mathrm{.}$

The integral exists by A continuous function on $[a,b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion and is the Darboux integral of The lower and upper Darboux integrals of a bounded $f$ on $[a,b]$ as ${\sup }_{P}L(f,P)$ and ${\inf }_{P}U(f,P)$, Darboux integrability as their equality, and the notation ${\int }_a^{b}f$. The square map is continuous by Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function and strictly increasing on the nonnegative reals by Monotonicity of $x\mapsto x^n$ and of $n\mapsto a^n$, so Continuous inverse theorem: a continuous injective $f$ on an interval $I$ is a bijection onto the order-convex set $f[I]$, and the inverse $g:f[I]\to I$ is continuous and strictly monotone in the same sense as $f$ makes its inverse square-root function continuous; existence and uniqueness are Square roots exist: a unique $\sqrt{a}\ge 0$ with $(\sqrt{a}{)}^2=a$; the positives are $\{x^2:x\mathrm{\ne }0\}$. Hence, for $r>0$, the functions $x\mapsto \pm \sqrt{r^2-x^2}$ are well defined and continuous on $[-r,r]$, and the closed disc of radius $r$ is the region between them.

This is a local convention for regions between continuous graphs. It does not assign an area to an arbitrary bounded planar set.

For a positive real $p$, the following are equivalent:

1. $p=\pi$;
2. $p$ is the least positive zero of sine;
3. $p\mathrm{/}2$ is the least positive zero of cosine;
4. $2p$ is the least positive common period of sine and cosine.

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

For every centre $c\in {\mathbb{R}}^2$ and radius $r>0$, the once-traversed circle has circumference

$C(r)=2\pi r\mathrm{.}$

Since its diameter is $d(r)=2r$, one has $C(r)\mathrm{/}d(r)=\pi$.

Let

$\gamma (t)=(\cos t,\sin t),0\le t\le 2\pi \mathrm{.}$

For $0<t\le 2\pi$ the restriction $\gamma \text{\hspace{0.17em}⁣}{\upharpoonright }_{[0,t]}$ is a circular arc of the unit circle (Circular arcs, circumference as arc length, and diameter), and the counterclockwise angle swept from $\gamma (0)=(1,0)$ to $\gamma (t)$ is defined to have radian measure

$L\left(\gamma \text{\hspace{0.17em}⁣}{\upharpoonright }_{[0,t]}\right),$

the length of that arc. At $t=0$ nothing is swept, and $\gamma \text{\hspace{0.17em}⁣}{\upharpoonright }_{[0,0]}$ is not a circular arc, that definition admitting only parameter intervals $[\alpha ,\beta ]$ with $\alpha <\beta$; it is the one-point path at $(1,0)$, whose length is $0$ by the singleton convention for path length (Paths in ${\mathbb{R}}^n$, inscribed polygonal sums, arc length as their supremum, and rectifiability), and the degenerate angle at $t=0$ is defined to have radian measure $0$. In every case, then, the radian measure of the swept angle is $L(\gamma \text{\hspace{0.17em}⁣}{\upharpoonright }_{[0,t]})$.

That measure is $t$. Fix $t$ with $0<t\le 2\pi$, and write $v(u):=(-\sin u,\cos u)$ for $u\in [0,t]$.

Sine and cosine are differentiable on $\mathbb{R}$ with $(\sin u{)}^{\mathrm{\prime }}=\cos u$ and $(\cos u{)}^{\mathrm{\prime }}=-\sin u$ (The derivatives of sine and cosine are cosine and minus sine), hence continuous on $\mathbb{R}$ (A function differentiable at $c$ is continuous at $c$), and so is $-\sin$, a scalar multiple of a continuous function (Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function, clause 1). Continuity of a real function passes to a subset of its domain, the condition on the restriction quantifying over fewer points (Continuity of $f:A\to \mathbb{R}$ at a point of $A$ and on $A$: the $\epsilon$-$\delta$ condition, its agreement with ${\lim }_{x\to c}f(x)=f(c)$ at a limit point, and continuity at an isolated point), so $\cos$, $\sin$ and $-\sin$ restricted to $[0,t]$ are continuous at every point of $[0,t]$; and for a real function on a subset of $\mathbb{R}$ the $\mathbb{R}$-native and the metric-space notions of continuity are the same notion (Dictionary: for $A\subseteq \mathbb{R}$ with the metric $d(x,y)=\mathrm{\mid }x-y\mathrm{\mid }$, continuity and uniform continuity of $f:A\to \mathbb{R}$ agree with the metric-space notions, the Lipschitz and Hölder conditions are the metric ones instantiated, and a subset of $\mathbb{R}$ is compact in the open-cover sense of $\mathbb{R}$ exactly when it is a compact metric subspace, clause 1). A function into ${\mathbb{R}}^m$ is continuous at a point of its domain if and only if each of its components is (A vector-valued function has a limit, or is continuous, if and only if each of its components does; with the algebra of continuous vector-valued functions, clause 1; Vector-valued functions $f:A\to {\mathbb{R}}^m$, their limits and continuity, with the dictionary to the metric notions). Hence $\gamma \text{\hspace{0.17em}⁣}{\upharpoonright }_{[0,t]}$ is continuous on $[0,t]$, and so is $v:[0,t]\to {\mathbb{R}}^2$.

Every point of a nondegenerate interval is a limit point of it, and if a real function is differentiable at a point of its domain, so is its restriction to any subset still having that point as a limit point, with the same derivative (The derivative $f^{\mathrm{\prime }}(c)={\lim }_{x\to c}\frac{f(x)-f(c)}{x-c}$ of $f:A\to \mathbb{R}$ at a point $c\in A$ that is a limit point of $A$, and differentiability on a set). So $\cos$ and $\sin$ restricted to $[0,t]$ are differentiable at every $u\in (0,t)$, with derivatives $-\sin u$ and $\cos u$; and a vector-valued function is differentiable at a limit point of its domain exactly when each component is, its derivative there being the vector of the component derivatives (The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral). Hence $\gamma \text{\hspace{0.17em}⁣}{\upharpoonright }_{[0,t]}$ is differentiable at every $u\in (0,t)$ with derivative $(-\sin u,\cos u)=v(u)$, and $v$ is a continuous extension of that derivative to $[0,t]$ — the hypotheses of If $\gamma :[a,b]\to {\mathbb{R}}^n$ is continuous, differentiable on $(a,b)$, and ${\gamma }^{\mathrm{\prime }}$ extends continuously to $[a,b]$, then $L(\gamma )={\int }_a^b\parallel {\gamma }^{\mathrm{\prime }}(t){\parallel }_{2}dt$. Since $\parallel w{\parallel }_2=(\mathrm{\mid }{w}_0{\mathrm{\mid }}^2+\mathrm{\mid }{w}_1{\mathrm{\mid }}^2{)}^{1\mathrm{/}2}$ (The $p$-norms $\parallel x{\parallel }_p$ for rational $p\ge 1$, and $\parallel x{\parallel }_{\mathrm{\infty }}$) and ${\sin }^{2}u+{\cos }^{2}u=1$ (Parity and the Pythagorean identity for sine and cosine), we get $\parallel v(u){\parallel }_2=(\mathrm{\mid }-\sin u{\mathrm{\mid }}^2+\mathrm{\mid }\cos u{\mathrm{\mid }}^2{)}^{1\mathrm{/}2}=({\sin }^{2}u+{\cos }^{2}u{)}^{1\mathrm{/}2}=1$ for every $u\in [0,t]$; and the integral of a constant over $[0,t]$ is that constant times $t$ (If $m\le f\le M$ on $[a,b]$ then $m(b-a)\le L(f,P)\le \underset{‾}{{\int }_a^b}f\le \overline{{\int }_a^b}f\le U(f,P)\le M(b-a)$ for every partition $P$; in particular every constant function is integrable, with ${\int }_a^{b}c=c(b-a)$). Therefore

$L\left(\gamma \text{\hspace{0.17em}⁣}{\upharpoonright }_{[0,t]}\right)={\int }_0^t\parallel v(u){\parallel }_{2}du={\int }_0^{t}1du=t$

(If $\gamma :[a,b]\to {\mathbb{R}}^n$ is continuous, differentiable on $(a,b)$, and ${\gamma }^{\mathrm{\prime }}$ extends continuously to $[a,b]$, then $L(\gamma )={\int }_a^b\parallel {\gamma }^{\mathrm{\prime }}(t){\parallel }_{2}dt$), while at $t=0$ both the length and the parameter are $0$.

Thus the analytic parameter $t$ is the geometric radian measure of the swept angle. At $t=2\pi$ the path makes one full turn, so a full turn has radian measure $2\pi$, agreeing with the circumference of the unit circle (Every circle has circumference 2 pi r and circumference-to-diameter ratio pi).

Given $A,O>0$, put

$H=\sqrt{A^2+O^2},P=(A,O)\mathrm{.}$

There is a unique $\theta \in (0,\pi \mathrm{/}2)$ such that

$(\frac{A}{H},\frac{O}{H})=(\cos \theta ,\sin \theta )\mathrm{.}$

The counterclockwise unit-circle arc from $(1,0)$ to $(A\mathrm{/}H,O\mathrm{/}H)$ has radian measure $\theta$. The coordinate right triangle with vertices $(0,0)$, $(A,0)$, and $P$ therefore satisfies

$\cos \theta =\frac{A}{H},\sin \theta =\frac{O}{H}\mathrm{.}$

The strict hypotheses $A,O>0$ make the triangle nondegenerate and place $\theta$ in the acute range $(0,\pi \mathrm{/}2)$. If one of the legs is zero, the normalized point lies on a coordinate axis and the radian definition still supplies the corresponding unit-circle value, but the resulting configuration is not a nondegenerate right triangle; the theorem does not impose an acute-triangle side-ratio convention on those axis or quadrantal cases.

What is measured, and what is not. The library assigns radian measure only to a counterclockwise unit-circle arc starting at $(1,0)$ (Radian angle by unit-circle arc length); it defines no interior angle of a triangle, and no invariance of an angle under scaling. So the theorem identifies $\theta$ as the radian measure of the arc ending at $(A\mathrm{/}H,O\mathrm{/}H)$ — the unit-circle point of which the hypotenuse vertex $P=H(\cos \theta ,\sin \theta )$ is the positive multiple $H$ — and asserts the two side ratios of the triangle. It does not assert that the triangle's interior angle at the origin equals $\theta$, which would need a notion of angle the library has not built.

For every natural $n\ge 3$, let $I_n$ and $O_n$ be the perimeters of the regular $n$-gons respectively inscribed in and circumscribed about the unit circle. Then

$I_n=2n\sin (\pi \mathrm{/}n),O_n=2n\tan (\pi \mathrm{/}n),$

$I_n<2\pi <O_n\mathrm{.}$

The sequence $(I_n{)}_{n\ge 3}$ is strictly increasing, $(O_n{)}_{n\ge 3}$ is strictly decreasing, and both converge to $2\pi$, the circumference of the unit circle.

For every $r>0$, the Riemann area of the closed disc of radius $r$ is $\pi r^2$. In particular, the unit disc has area $\pi$.

The series

${\sum }_{k=0}^{\mathrm{\infty }}\frac{(-1{)}^k}{2k+1}$

converges, and its sum is $\pi \mathrm{/}4$. More precisely, for every natural $N$,

$\frac{\pi }{4}={\sum }_{k=0}^N\frac{(-1{)}^k}{2k+1}+R_N,\mathrm{\mid }{R}_N\mathrm{\mid }\le \frac{1}{2N+3},$

where

$R_N:=(-1{)}^{N+1}{\int }_0^1\frac{x^{2N+2}}{1+x^2}dx\mathrm{.}$