How can I draw a sphere knowing that centre and radius?

In the warped coordinate system, if we draw a circle at O of radius 1 it will pass through points A, B, C and D. However, the point of maximum distance (screen coordinates) occurs at point R. Therefore the outer edge of a sphere will also pass though point R.

Using the screen coordinates for R and O one can compute this radius. From the screen coordinates of A and O one can compute the y radius of an ellipse passing through A.

\documentclass[12pt,border=3]{standalone}
\usepackage{tikz}
\usepackage{tikz-3dplot}
\usepackage{fouriernc}
\usetikzlibrary{calc}

\newlength{\radius}

\newcommand{\RightAngle}[4][2.5pt]{%
        \draw ($#3!#1!#2$)
        --($ #3!2!($($#3!#1!#2$)!.5!($#3!#1!#4$)$) $)
        --($#3!#1!#4$) ;
        }
\begin{document}

\newcommand\pgfmathsinandcos[3]{%
  \pgfmathsetmacro#1{sin(#3)}%
  \pgfmathsetmacro#2{cos(#3)}%
}

\begin{tikzpicture}[scale=2.5]

\pgfmathsetmacro\AngleFuite{135}
\pgfmathsetmacro\coeffReduc{.85}
\pgfmathsetmacro\clen{2}
\pgfmathsinandcos\sint\cost{\AngleFuite}
\begin{scope} [x     = {(\coeffReduc*\cost,-\coeffReduc*\sint)},
               y     = {(1cm,0cm)},
               z     = {(0cm,1cm)}]

\path
      coordinate (A) at (0,0,0)
      coordinate (B) at (1,0,0)
      coordinate (C) at (1,{sqrt(3)},0)
      coordinate (D) at (0,{sqrt(3)},0)
      coordinate (S) at (0,0,2)
      coordinate(H) at (4/5,0,2/5)
      coordinate(K) at (0,{4*sqrt(3)/7},6/7)
      coordinate(O) at (1/2,{sqrt(3)/2},0)
            ;
            \draw[thick] (C)--(K)
                        ;
      \draw[ dashed] (A)--(B) (A)--(D) (A)--(K)
      (A)--(S) (A)--(H)
      (A)--(C)
      (B)--(D) ;
  \pgfmathsetmacro{\angle}{0.5*atan2(2*\coeffReduc*\cost, \coeffReduc*\coeffReduc-1)}% find max distance
  %\draw[red] (O) +(\angle:-1) -- +(\angle:1);% angle test
  \path (O) ++(\angle:1) coordinate (R);
  \draw[red] (R) arc[radius=1,start angle={\angle},end angle={\angle+180}];
  \draw[red,dashed] (R) arc[radius=1,start angle={\angle},end angle={\angle-180}];
  \end{scope}
% comput radius of circle passing through (B)
\pgfpointdiff{\pgfpointanchor{R}{center}}{\pgfpointanchor{O}{center}}%
\pgfgetlastxy{\xRO}{\yRO}% \xRO = radius cos(angle), yBO = radius sin(angle)
\pgfmathsetmacro{\angle}{atan(\yRO/\xRO)}%
\pgfmathsetlength{\radius}{\xRO/cos(\angle)}% with units
\draw (O) circle[radius={\radius}];
% compute radius of ellipse passing through (A)
\pgfpointdiff{\pgfpointanchor{A}{center}}{\pgfpointanchor{O}{center}}%
\pgfgetlastxy{\xAO}{\yAO}% \xAO = x radius * cos(angle), \yAO= y radius * sin(angle)
\pgfmathsetmacro{\angle}{acos(\xAO/\radius)};
\draw (O) ++(0:\radius) arc[x radius={\radius}, y radius={\yAO/sin(\angle)}, start angle=0, end angle=180];
\draw[dashed] (O) ++(0:\radius) arc[x radius={\radius}, y radius={\yAO/sin(\angle)}, start angle=0, end angle=-180];
%
  \RightAngle{(C)}{(K)}{(A)};
  \RightAngle{(S)}{(H)}{(A)};
\draw [fill opacity=0.4,fill=green!80!blue] (S) -- (B) -- (C) -- cycle;
\draw [fill opacity=0.4,fill=orange!80!blue] (S) -- (C) -- (D) -- cycle;
\shade[ball color=blue, opacity=.5] (O) circle (\radius);% order of shaning is important
\foreach \v/\position in {A/below,B/below,C/below,D/right,S/above,H/left,K/right,O/below} {
    \draw[fill=black] (\v) circle (0.7pt) node [\position=0.2mm] {$\v$};
}

\end{tikzpicture}
\end{document} 

I used a translation to get the point O as the origin. I got the same result. My code

\documentclass[border=2mm,12pt]{standalone}
\usepackage{tikz}
\usepackage{tkz-euclide}
\usetkzobj{all}
\usepackage{tikz-3dplot}
\usetikzlibrary{calc,backgrounds}
%\usetikzlibrary{arrows,calc}
\begin{document}
\tdplotsetmaincoords{60}{110}
\begin{tikzpicture}[scale=2]
\def\R{1}
\coordinate (O) at (0,0,0);
\fill[ball color=cyan!90, opacity=1.0]  (O) circle (\R); % 3D lighting effect
 \begin{scope}[tdplot_main_coords, shift={(0,0)}, rotate=0]
\path
coordinate (A) at (-1/2,-{sqrt(3)/2},0)
      coordinate (B) at (1/2,-{sqrt(3)/2},0)
      coordinate (H) at (3/10,-{sqrt(3)/2},2/5)
      coordinate (K) at (-1/2,{sqrt(3)/14},6/7)
       coordinate (C) at (1/2,{sqrt(3)/2},0)
            coordinate (D) at (-1/2,{sqrt(3)/2},0)
      coordinate (S) at (-1/2,-{sqrt(3)/2},2);
\draw[ dashed]
(A)--(S) (A)--(B) (A)--(D) (A)--(H) (A)--(K) (A)--(C) (B)--(D) (S)--(O) (H)--(C) (K)--(C)
       ;
\draw [fill opacity=0.4,fill=green!80!blue] (S) -- (B) -- (C) -- cycle;
\draw [fill opacity=0.4,fill=orange!80!blue] (S) -- (C) -- (D) -- cycle;
\foreach \v/\position in {A/below,B/left,C/below,D/right,S/above,O/{above right},H/left,K/right} {
    \draw[fill=black] (\v) circle (0.7pt) node [\position=0.2mm] {$\v$};
}
\end{scope}
\tkzMarkRightAngle[size = 0.1](A,H,C);
\tkzMarkRightAngle[size = 0.1](A,K,C);
\end{tikzpicture}
\end{document} 

Another code

\documentclass[border=2mm,12pt]{standalone}
    \usepackage{tikz}
    \usepackage{tkz-euclide}
    \usetkzobj{all}
    \usepackage{tikz-3dplot}
    \usepackage{fouriernc}
    \usetikzlibrary{calc,backgrounds}
     \usetikzlibrary{intersections,calc,backgrounds}
    \begin{document}
        \tdplotsetmaincoords{60}{120}
        \begin{tikzpicture}[tdplot_main_coords,scale=1.5]
        \pgfmathsetmacro\a{3}
        \pgfmathsetmacro\b{4}
        \pgfmathsetmacro\h{5}
        \pgfmathsetmacro\r{sqrt(\a^2 + \b^2)/2} 

        % definitions
        \path 
        coordinate(A) at (0,0,0)
        coordinate (B) at (\a,0,0)
        coordinate (C) at (\a,\b,0) 
        coordinate (D) at (0,\b,0)                            
        coordinate (S) at (0,0,\h)
        coordinate (E) at ({\a*\h^2/(\a^2+\h^2)}, 0, {\a^2*\h/(\a^2+\h^2)})
        coordinate (F) at (0, {\b*\h^2/(\b^2+\h^2)}, {\b^2*\h/(\b^2+\h^2)})
        coordinate (K) at ({\a*\h^2/(\a^2+\b^2+\h^2)}, {\b*\h^2/(\a^2+\b^2+\h^2)}, {(\a^2+\b^2)*\h/(\a^2+\b^2+\h^2)});              

        \begin{scope}
        \draw [dashed, thick, name path=B--D] (B) -- (D);
        \draw [dashed, thick, name path=C--A] (C) -- (A);
        \path [name intersections={of=B--D and C--A,by=O}];
        \end{scope}

        \begin{scope}
        \draw [dashed, thick, name path=S--O] (S) -- (O);
        \draw [dashed, thick, name path=E--F] (E) -- (F);
        \path [name intersections={of=S--O and E--F,by=I}];
        \end{scope}

        \begin{scope}
        \draw[dashed, thick]
        (A) -- (B)   (D)--(A) (S)--(A);
        \draw[dashed, thick]
        (E) --(A)  -- (F);
        \draw[ultra thick]
        (S) -- (B) -- (C) -- (D)--cycle (S)--(C) (E) -- (K) --(F);
        \draw [thick, dashed] (A) -- (K)  (E) -- (F) ;
        \tkzMarkRightAngle(S,A,D)
        \tkzMarkRightAngle(S,A,B)
        \tkzMarkRightAngle(A,B,C)
        \tkzMarkRightAngle(B,A,D)
        \tkzMarkRightAngle(A,F,D)
        \tkzMarkRightAngle(A,E,B)
        \tkzMarkRightAngle(A,K,C)
        \end{scope}

        \begin{scope}[tdplot_screen_coords, on background layer]
        \fill[ball color=cyan!90, opacity=1.0] (O) circle (\r); % 3D lighting effect
        \end{scope}

        \foreach \point/\position in {A/left,B/below,C/below,S/above,D/right,E/left,D/right,F/right,K/above right,O/below,I/below}
        {
            \fill (\point) circle (1.5pt);
            \node[\position=3pt] at (\point) {$\point$};
        }
        \end{tikzpicture}
    \end{document}