Circle Area and Perimeter

$\begin{tikzpicture} % grid \draw[help lines] (-2,-2) grid (2,2); % origin \draw[red, line width=.1mm] (-0.1,-0.1) -- (0.1,0.1) (0.1,-0.1) -- (-0.1,0.1); \coordinate[label={[red]above left:$O$}] (O) at (0,0); \coordinate[label={[red]above right:$R$}] (R) at (2,0); \coordinate[label={[red]above right:$Q$}] (Q) at (0,2); \coordinate[label={[red]above right:$C$}] (C) at (2,2); %radius \draw[fill=blue!20,fill opacity=0.1] (O) -- (R) -- (C) -- (Q) -- cycle; % triangle \draw [black] (0,0) circle [radius=2]; \drawbrace{O}{R}{2mm}{blue}{$r$}{0}{-4mm}{mirror}; \node [thick] at (1,1) {$A_{\square}=r^{2}$}; \node [thick] at (0,-1) {$A_{\bullet}=\pi*r^{2}$}; \node [thick] at (0,-1.5) {$P_{\bullet}=2*\pi*r$}; \end{tikzpicture}$

A circle contains the maximal surface in relation to its perimeter. This translates to spheres as well, since a sphere contains the maximal volume in relation to its surface.

Equation of a Circle

$\begin{tikzpicture} % grid \draw[help lines] (-2,-2) grid (2,2); % origin \draw[red, line width=.1mm] (-0.1,-0.1) -- (0.1,0.1) (0.1,-0.1) -- (-0.1,0.1); \coordinate[label={[red]above left:$O$}] (O) at (0,0); \coordinate[label={[red]above right:$R$}] (R) at (2,0); \coordinate[label={[red]above right:$Q$}] (Q) at ({cos(30)},{2*cos(30)}); %projection \draw[black,dotted] (Q) -- ({cos(30)},0); \node [square,minimum size=1mm,dotted] at ({cos(30)-0.1},0.1) [draw] (d2) [black] {}; \drawpoint{Q}{.5mm}{red}; %radius %\draw[fill=blue!20,fill opacity=0.1] (O) -- (R); % circle \draw [black] (0,0) circle [radius=2]; \draw [black] (O) -- (R); \draw [black] (O) -- (Q); \drawbrace{O}{R}{2mm}{blue}{$r$}{0}{-4mm}{mirror}; \drawbrace{O}{Q}{2mm}{blue}{$r$}{-4mm}{0}{}; \end{tikzpicture}$

$\begin{eqnarray*} (x-x_{0})^{2}+(y-y_{0})^{2} &=& r^{2} \end{eqnarray*}$

where the parameters $x_{0}$ and $y_{0}$ represent an offset from the origin point $O$ (usually both equal to zero if the circle is centred in the origin point).