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).


fuss/mathematics/geometry/shapes/circles.txt ยท Last modified: 2017/02/22 18:30 (external edit)

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