finding coordinates to the end point of an arc

You can construct the coordinate C first and then \filldraw the whole figure in one sweep.

\documentclass[border=2pt]{standalone}
\usepackage{tikz}
\begin{document}
\begin{tikzpicture}
  \path
     (9.7,-3) coordinate(D)
     arc [radius=4.7, start angle=360, end angle=315] coordinate(C);
  \filldraw[fill=red, draw=red]
     (C)
     arc [radius=4.7, start angle=315, end angle=360]
     -- (0.3,-3) coordinate(A)
     arc [radius=4.7, start angle=180, end angle=225] coordinate(B);
\end{tikzpicture}
\end{document}

Or you can compute the coordinates of (C).

\documentclass[border=2pt]{standalone}
\usepackage{tikz}
\begin{document}
\begin{tikzpicture}
  \newcommand\R{4.7}% radius
  \newcommand\A{45}% angle
  \filldraw[fill=red, draw=red]
     (0.3,-3) coordinate (A)
     arc [radius=\R, start angle=180, end angle={180+\A}] coordinate (B)
     -- ({5+\R*cos(\A)},{-3-\R*sin(\A)}) coordinate (C)
     arc [radius=\R, start angle={360-\A}, end angle=360] coordinate (D);
\end{tikzpicture}
\end{document}

Or you can let tikz compute the coordinates.

\documentclass[border=2pt]{standalone}
\usepackage{tikz}
\usetikzlibrary{calc}
\begin{document}
\begin{tikzpicture}
  \newcommand\R{4.7}% radius
  \newcommand\A{45}% angle
  \coordinate (M) at (5,-3);
  \coordinate (A) at ($(M)+(180:\R)$);
  \coordinate (B) at ($(M)+({180+\A}:\R)$);
  \coordinate (C) at ($(M)+({360-\A}:\R)$);
  \coordinate (D) at ($(M)+(360:\R)$);
  \filldraw[fill=red, draw=red]
     (A) arc [radius=\R, start angle=180, end angle={180+\A}]
     -- (C) arc [radius=\R, start angle={360-\A}, end angle=360];
\end{tikzpicture}
\end{document}

My trigonometry knowledge doesn't allow me to compute the height of the figure, but if you know how to do it, you can use it to clip a circle:

\documentclass[11pt]{article}
\usepackage[a4paper, margin=1in]{geometry}
\usepackage{tikz}

\begin{document}
\begin{tikzpicture}

\begin{scope}
    \clip (0.3,-3) rectangle (9.7,-6.34); 
    \fill[red] (5,-3) circle(4.7);
\end{scope}
\end{tikzpicture}
\end{document}

Update 1:

gernot refreshed my trigonometry and provided the solution:

\documentclass[11pt]{article}
\usepackage[a4paper, margin=1in]{geometry}
\usepackage{tikz}

\begin{document}
\begin{tikzpicture}

\begin{scope}
    %radius*sin(225-180) = 4.7/sqrt(2) = 3,3234018
    \clip (0.3,-3) rectangle (9.7,-6.323); 
    \fill[red] (5,-3) circle(4.7);
\end{scope}
\end{tikzpicture}
\end{document}

Update 2:

And StefanH lets to TikZ calc library to compute the height for us:

\documentclass[11pt]{article}
\usepackage[a4paper, margin=1in]{geometry}
\usepackage{tikz}
\usetikzlibrary{calc}  %<---- Load calc library

\begin{document}
\begin{tikzpicture}

\begin{scope}
    \clip (0.3,-3) rectangle ({(9.7,-3)} |- {$(5,-3)+(315:4.7)$}); 
    \fill[red] (5,-3) circle(4.7);
\end{scope}
\end{tikzpicture}
\end{document}