Table of Contents

Equation of a Sphere

\begin{tikzpicture} % MERC %% some definitions \def\R{3} % sphere radius \def\angEl{25} % elevation angle \def\angAz{-100} % azimuth angle \def\angPhiOne{-50} % longitude of point P \def\angPhiTwo{-35} % longitude of point Q \def\angBeta{33} % latitude of point P and Q %% working planes \pgfmathsetmacro\H{\R*cos(\angEl)} % distance to north pole \LongitudePlane[xzplane]{\angEl}{\angAz} \LongitudePlane[pzplane]{\angEl}{\angPhiOne} \LongitudePlane[qzplane]{\angEl}{\angPhiTwo} \LatitudePlane[equator]{\angEl}{0} %% draw background sphere \fill[ball color=grey!10] (0,0) circle (\R); % 3D lighting effect %% characteristic points \path[pzplane] (\angBeta:\R) coordinate [label={[black]above left:$P$}] (P); \drawpoint{P}{.5mm}{black}; \coordinate [label={[black]above left:$O$}] (O) at (0,0); \drawpoint{O}{.5mm}{black}; \path[pzplane] (\R,0) coordinate (PE); \path[xzplane] (\R,0) coordinate (XE); \path[pzplane] (\R,0) coordinate [label={[black]above right:$R$}] (R); \drawpoint{R}{.5mm}{black}; \draw[dotted,black] (O) -- (R); \drawbrace{O}{R}{2mm}{black}{$r$}{0}{-4mm}{mirror}; %% meridians and latitude circles \DrawLatitudeCircle[\R]{0} % equator %% draw lines and put labels %\draw (-\R,-\H) -- (-\R,2*\R) (\R,-\H) -- (\R,2*\R); \draw[->] (O) -- (P); % z axis \draw[->,dashed,blue] (O) -- +(0,1.5*\R) node[above] {$z$}; % x axis \path[xzplane] (0:\R) node[below,red] {$x$} coordinate (x); \draw[->,dashed,red] (O) -- (x); % y axis \path[pyplane] (0:\R) node[left,green!50] {$y$} coordinate (y); \draw[->,dashed,green!50] (O) -- (y); % projection P \path[pzplane] (0.8*\R,0) coordinate [label={[black]above right:$Q$}] (Q); \drawpoint{Q}{.5mm}{black}; \draw[->,dotted] (P) -- (Q); %angles \draw[pzplane,->,thin] (0:0.5*\R) to[bend right=15] node[midway,right] {$\alpha$} (\angBeta:0.5*\R); \draw[equator,->,thin] (\angAz:0.5*\R) to[bend right=30] node[pos=0.4,below] {$\beta$} (\angPhiOne:0.5*\R); \end{tikzpicture}

\begin{eqnarray*} x &=& r * \cos{\alpha} * \cos{\beta} & \\ y &=& r * \cos{\alpha} * \sin{\beta} & \\ z &=& r * \sin{\alpha} \end{eqnarray*}

Volume of a Sphere

\begin{tikzpicture} %% some definitions \def\R{3} % sphere radius \def\angEl{25} % elevation angle \def\angAz{-100} % azimuth angle \def\angPhiOne{-50} % longitude of point P \def\angPhiTwo{-35} % longitude of point Q \def\angBeta{33} % latitude of point P and Q %% working planes \pgfmathsetmacro\H{\R*cos(\angEl)} % distance to north pole \LongitudePlane[xzplane]{\angEl}{\angAz} \LongitudePlane[pzplane]{\angEl}{\angPhiOne} \LongitudePlane[qzplane]{\angEl}{\angPhiTwo} \LatitudePlane[equator]{\angEl}{0} %% draw background sphere \fill[ball color=grey!10] (0,0) circle (\R); % 3D lighting effect % circles \foreach \t in {-80,-60,...,80} { \DrawLatitudeCircle[\R]{\t} } %\foreach \t in {-5,-35,...,-175} { \DrawLongitudeCircle[\R]{\t} } % z axis \draw[->,dashed,blue] (O) -- +(0,\R) node[above] {$z$} coordinate (z); % x axis \path[xzplane] (0:\R) node[below,red] {$x$} coordinate (x); \draw[->,dashed,red] (O) -- (x); % y axis \path[pyplane] (0:\R) node[left,green!50] {$y$} coordinate (y); \draw[->,dashed,green!50] (O) -- (y); % Ox %\drawbrace{O}{x}{2mm}{black}{$r$}{4mm}{0}{}; % Oy %\drawbrace{O}{y}{2mm}{black}{$r$}{0}{4mm}{}; % Oz %\drawbrace{O}{z}{2mm}{black}{$r$}{-4mm}{0}{}; % right-angle %\node[square,minimum size=1mm, dotted] at (0.1,0.1) [draw,fill] (O) [magenta] {}; %\coordinate[label={[magenta]above right:$90^{\circ}$}] (90) at (0.2, 0.2); \coordinate [label={[black]left:$O$}] (O) at (0,0); \drawpoint{O}{.5mm}{black}; \path[pzplane] (\angBeta:\R) coordinate [label={[black]right:$P$}] (P); \drawpoint{P}{.5mm}{black}; \draw[->,black] (O) -- (P); \drawbrace{O}{P}{2mm}{black}{$r$}{0}{-4mm}{mirror}; \coordinate [label={[black]left:$Q$}] (Q) at (0,0.5*\R); \drawpoint{Q}{.5mm}{black}; \draw[dotted] (P) -- (Q); \end{tikzpicture}

\begin{eqnarray*} V &=& \frac{4}{3}\pi*r^{3} \end{eqnarray*}

Area of a Sphere

\begin{tikzpicture} %% some definitions \def\R{3} % sphere radius \def\angEl{25} % elevation angle \def\angAz{-100} % azimuth angle \def\angPhiOne{-50} % longitude of point P \def\angPhiTwo{-35} % longitude of point Q \def\angBeta{33} % latitude of point P and Q %% working planes \pgfmathsetmacro\H{\R*cos(\angEl)} % distance to north pole \LongitudePlane[xzplane]{\angEl}{\angAz} \LongitudePlane[pzplane]{\angEl}{\angPhiOne} \LongitudePlane[qzplane]{\angEl}{\angPhiTwo} \LatitudePlane[equator]{\angEl}{0} %% draw background sphere \fill[ball color=gray!10] (0,0) circle (\R); % 3D lighting effect \fill[ball color=gray!75] (0,0) circle (0.75*\R); % 3D lighting effect % circles %\foreach \t in {-80,-60,...,80} { \DrawLatitudeCircle[\R]{\t} } % z axis \draw[->,dashed,blue] (O) -- +(0,\R) node[above] {$z$} coordinate (z); % x axis \path[xzplane] (0:\R) node[below,red] {$x$} coordinate (x); \draw[->,dashed,red] (O) -- (x); % y axis \path[pyplane] (0:\R) node[left,green!50] {$y$} coordinate (y); \draw[->,dashed,green!50] (O) -- (y); % Ox %\drawbrace{O}{x}{2mm}{black}{$r$}{4mm}{0}{}; % Oy %\drawbrace{O}{y}{2mm}{black}{$r$}{0}{4mm}{}; % Oz %\drawbrace{O}{z}{2mm}{black}{$r$}{-4mm}{0}{}; % right-angle %\node[square,minimum size=1mm, dotted] at (0.1,0.1) [draw,fill] (O) [magenta] {}; %\coordinate[label={[magenta]above right:$90^{\circ}$}] (90) at (0.2, 0.2); \coordinate [label={[black]left:$O$}] (O) at (0,0); \drawpoint{O}{.5mm}{black}; \path[pzplane] (\angBeta:\R) coordinate [label={[black]right:$P$}] (P); \drawpoint{P}{.5mm}{black}; \draw[->,black] (O) -- (P); \drawbrace{O}{P}{2mm}{black}{$r$}{0}{-4mm}{mirror}; \coordinate [label={[black]left:$Q$}] (Q) at (0,0.5*\R); \drawpoint{Q}{.5mm}{black}; \draw[dotted] (P) -- (Q); \end{tikzpicture}

\begin{eqnarray*} A &=& 4\pi r^{2} \end{eqnarray*}