Problem source

\begin{center}
\begin{tikzpicture}
% Setting up the same viewport/dimensions
\clip (-0.45,1.84) rectangle (8.6,7.17);
% Creating the filled polygons with opacity
\filldraw[color=black, opacity=0.05] 
  (1.82,3.17) -- (1.91,3.21) -- (2.04,3.27) -- (2.16,3.32) -- (2.28,3.36) -- (2.45,3.42) -- 
  (2.58,3.45) -- (2.75,3.48) -- (2.93,3.51) -- (3.06,3.52) -- (3.24,3.53) -- (3.33,3.53) -- 
  (3.23,3.54) -- (3.13,3.57) -- (2.99,3.62) -- (2.88,3.67) -- (2.76,3.75) -- (2.66,3.84) -- 
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  (2.02,3.3) -- (1.93,3.23) -- cycle;
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  (4.51,4.24) -- (4.49,4.1) -- (4.48,3.99) -- (4.49,3.88) -- (4.5,3.76) -- (4.53,3.65) -- 
  (4.58,3.52) -- (4.64,3.4) -- (4.7,3.3) -- (4.78,3.2) -- (4.85,3.12) -- (5.05,2.96) -- 
  (4.83,3.1) -- (4.71,3.17) -- (4.51,3.26) -- (4.34,3.33) -- (4.19,3.39) -- (4.03,3.43) -- 
  (3.89,3.46) -- (3.76,3.49) -- (3.67,3.5) -- (3.62,3.51) -- (3.4,3.52) -- (3.52,3.52) -- 
  (3.68,3.54) -- (3.82,3.58) -- (3.95,3.64) -- (4.08,3.71) -- (4.18,3.79) -- (4.28,3.88) -- 
  (4.38,4) -- (4.45,4.12) -- cycle;
\filldraw[color=black, opacity=0.05] 
  (0.77,5.07) -- (0.94,5.17) -- (1.14,5.27) -- (1.3,5.34) -- (1.47,5.41) -- (1.59,5.46) -- 
  (1.74,5.51) -- (1.86,5.55) -- (2.02,5.6) -- (2.2,5.64) -- (2.35,5.68) -- (2.55,5.72) -- 
  (2.7,5.74) -- (2.84,5.76) -- (2.97,5.77) -- (3.13,5.79) -- (3.27,5.8) -- (3.44,5.8) -- 
  (3.34,5.79) -- (3.2,5.77) -- (3.09,5.74) -- (2.97,5.69) -- (2.87,5.64) -- (2.76,5.56) -- 
  (2.66,5.48) -- (2.58,5.39) -- (2.5,5.29) -- (2.43,5.17) -- (2.37,5.03) -- (2.34,4.9) -- 
  (2.32,4.77) -- (2.31,4.6) -- (2.34,4.41) -- (2.27,4.59) -- (2.21,4.7) -- (2.14,4.8) -- 
  (2.04,4.9) -- (1.93,5) -- (1.84,5.05) -- (1.75,5.1) -- (1.62,5.15) -- (1.51,5.18) -- 
  (1.37,5.2) -- (1.25,5.2) -- (1.16,5.2) -- (1.02,5.17) -- (0.89,5.13) -- cycle;
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  (6.31,5.02) -- (6.1,5.15) -- (5.92,5.24) -- (5.77,5.31) -- (5.61,5.38) -- (5.39,5.46) -- 
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  (3.8,5.79) -- (3.6,5.8) -- (3.51,5.79) -- (3.67,5.77) -- (3.79,5.74) -- (3.9,5.7) -- 
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  (4.56,4.9) -- (4.58,4.75) -- (4.59,4.59) -- (4.57,4.43) -- (4.63,4.57) -- (4.71,4.7) -- 
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  (5.57,5.19) -- (5.69,5.2) -- (5.82,5.19) -- (5.97,5.17) -- (6.15,5.11) -- cycle;
% Draw the curved lines (parametric plots)
\draw (9,0.3) arc (0:180:5.5); % First parametric plot
\draw (6.35,0.45) arc (0:180:3.08); % Second parametric plot
% Circles
\draw (1.29,4.12) circle (1.09);
\draw (3.45,4.66) circle (1.14);
\draw (5.69,3.99) circle (1.21);
% Lines
\draw (1.29,4.12) -- (2.35,4.38);
\draw (3.45,4.66) -- (3.46,5.8);
\draw (5.69,3.99) -- (4.93,3.04);
% Arrows
\draw[->] (2.14,6.66) -- (2.16,5.1);
\draw[->] (4.72,6.7) -- (4.72,5.14);
\draw[->] (2.56,2.5) -- (2.56,3.74);
\draw[->] (4.4,2.42) -- (4.4,3.62);
% Text labels
\node at (1.9,7.1) [anchor=north west] {$X/n$};
\node at (4.5,7.1) [anchor=north west] {$X/n$};
\node at (2.34,2.43) [anchor=north west] {$Y/n$};
\node at (4.16,2.41) [anchor=north west] {$Y/n$};
\node at (3.7,5.2) [anchor=north west] {$r$};
\node at (5.54,3.53) [anchor=north west] {$r$};
\node at (1.74,4.1) [anchor=north west] {$r$};
\node at (7.55,4.43) [anchor=north west] {$R+r$};
\node at (5.95,2.41) [anchor=north west] {$R-r$};
\end{tikzpicture}
\par\end{center}
The region $A$ between concentric circles of radii $R+r$, $R-r$ contains $n$ circles of radius $r$. Each circle of radius $r$ touches both of the larger circles as well as its two neighbours of radius $r$, as shown in the figure. Find the relationship which must hold between $n,R$ and $r$. 
Show that $Y$, the total area of $A$ outside the circle of radius $r$ and adjacent to the circle of radius $R-r$, is given by 
\[
Y=nr\sqrt{R^{2}-r^{2}}-\pi(R-r)^{2}-n\pi r^{2}\left(\frac{1}{2}-\frac{1}{n}\right).
\]
Find similar expressions for $X$, the total area of $A$ outside the circles of radius $r$ and adjacent to the circle of radius $R+r$, and for $Z$, the total area inside the circle of radius $r$. 
What value does $(X+Y)/Z$ approach when $n$ becomes large?

Solution source

\begin{center}
\begin{tikzpicture}
% Setting up the same viewport/dimensions
\clip (-0.45,1.84) rectangle (8.6,7.17);
% Creating the filled polygons with opacity
\filldraw[color=black, opacity=0.05] 
  (1.82,3.17) -- (1.91,3.21) -- (2.04,3.27) -- (2.16,3.32) -- (2.28,3.36) -- (2.45,3.42) -- 
  (2.58,3.45) -- (2.75,3.48) -- (2.93,3.51) -- (3.06,3.52) -- (3.24,3.53) -- (3.33,3.53) -- 
  (3.23,3.54) -- (3.13,3.57) -- (2.99,3.62) -- (2.88,3.67) -- (2.76,3.75) -- (2.66,3.84) -- 
  (2.56,3.95) -- (2.47,4.08) -- (2.41,4.19) -- (2.35,4.35) -- (2.38,4.16) -- (2.38,4.02) -- 
  (2.36,3.92) -- (2.33,3.8) -- (2.3,3.71) -- (2.24,3.58) -- (2.18,3.49) -- (2.1,3.38) -- 
  (2.02,3.3) -- (1.93,3.23) -- cycle;
\filldraw[color=black, opacity=0.05] 
  (4.51,4.24) -- (4.49,4.1) -- (4.48,3.99) -- (4.49,3.88) -- (4.5,3.76) -- (4.53,3.65) -- 
  (4.58,3.52) -- (4.64,3.4) -- (4.7,3.3) -- (4.78,3.2) -- (4.85,3.12) -- (5.05,2.96) -- 
  (4.83,3.1) -- (4.71,3.17) -- (4.51,3.26) -- (4.34,3.33) -- (4.19,3.39) -- (4.03,3.43) -- 
  (3.89,3.46) -- (3.76,3.49) -- (3.67,3.5) -- (3.62,3.51) -- (3.4,3.52) -- (3.52,3.52) -- 
  (3.68,3.54) -- (3.82,3.58) -- (3.95,3.64) -- (4.08,3.71) -- (4.18,3.79) -- (4.28,3.88) -- 
  (4.38,4) -- (4.45,4.12) -- cycle;
\filldraw[color=black, opacity=0.05] 
  (0.77,5.07) -- (0.94,5.17) -- (1.14,5.27) -- (1.3,5.34) -- (1.47,5.41) -- (1.59,5.46) -- 
  (1.74,5.51) -- (1.86,5.55) -- (2.02,5.6) -- (2.2,5.64) -- (2.35,5.68) -- (2.55,5.72) -- 
  (2.7,5.74) -- (2.84,5.76) -- (2.97,5.77) -- (3.13,5.79) -- (3.27,5.8) -- (3.44,5.8) -- 
  (3.34,5.79) -- (3.2,5.77) -- (3.09,5.74) -- (2.97,5.69) -- (2.87,5.64) -- (2.76,5.56) -- 
  (2.66,5.48) -- (2.58,5.39) -- (2.5,5.29) -- (2.43,5.17) -- (2.37,5.03) -- (2.34,4.9) -- 
  (2.32,4.77) -- (2.31,4.6) -- (2.34,4.41) -- (2.27,4.59) -- (2.21,4.7) -- (2.14,4.8) -- 
  (2.04,4.9) -- (1.93,5) -- (1.84,5.05) -- (1.75,5.1) -- (1.62,5.15) -- (1.51,5.18) -- 
  (1.37,5.2) -- (1.25,5.2) -- (1.16,5.2) -- (1.02,5.17) -- (0.89,5.13) -- cycle;
\filldraw[color=black, opacity=0.05] 
  (6.31,5.02) -- (6.1,5.15) -- (5.92,5.24) -- (5.77,5.31) -- (5.61,5.38) -- (5.39,5.46) -- 
  (5.12,5.56) -- (4.88,5.62) -- (4.64,5.68) -- (4.49,5.71) -- (4.24,5.75) -- (4.02,5.78) -- 
  (3.8,5.79) -- (3.6,5.8) -- (3.51,5.79) -- (3.67,5.77) -- (3.79,5.74) -- (3.9,5.7) -- 
  (4.03,5.64) -- (4.13,5.57) -- (4.24,5.48) -- (4.36,5.34) -- (4.46,5.17) -- (4.52,5.05) -- 
  (4.56,4.9) -- (4.58,4.75) -- (4.59,4.59) -- (4.57,4.43) -- (4.63,4.57) -- (4.71,4.7) -- 
  (4.78,4.79) -- (4.9,4.9) -- (5.03,5) -- (5.16,5.08) -- (5.3,5.14) -- (5.43,5.17) -- 
  (5.57,5.19) -- (5.69,5.2) -- (5.82,5.19) -- (5.97,5.17) -- (6.15,5.11) -- cycle;
% Draw the curved lines (parametric plots)
\draw (9,0.3) arc (0:180:5.5); % First parametric plot
\draw (6.35,0.45) arc (0:180:3.08); % Second parametric plot
% Circles
\draw (1.29,4.12) circle (1.09);
\draw (3.45,4.66) circle (1.14);
\draw (5.69,3.99) circle (1.21);
% Lines
\draw (1.29,4.12) -- (2.35,4.38);
% \draw (3.45,4.66) -- (3.46,5.8);
\draw (5.69,3.99) -- (4.93,3.04);

\draw (1.29,4.12) -- (2.35,4.38) -- (3.45,4.66) -- (3.1,0) -- cycle;
% Arrows
% \draw[->] (2.14,6.66) -- (2.16,5.1);
% \draw[->] (4.72,6.7) -- (4.72,5.14);
% \draw[->] (2.56,2.5) -- (2.56,3.74);
% \draw[->] (4.4,2.42) -- (4.4,3.62);
% Text labels
% \node at (1.9,7.1) [anchor=north west] {$X/n$};
% \node at (4.5,7.1) [anchor=north west] {$X/n$};
% \node at (2.34,2.43) [anchor=north west] {$Y/n$};
% \node at (4.16,2.41) [anchor=north west] {$Y/n$};
% \node at (3.7,5.2) [anchor=north west] {$r$};
% \node at (5.54,3.53) [anchor=north west] {$r$};
\node at (1.74,4.1) [anchor=north west] {$r$};
\node at (7.55,4.43) [anchor=north west] {$R+r$};
\node at (5.95,2.41) [anchor=north west] {$R-r$};
\end{tikzpicture}
\par\end{center}

The shown isoceles triangle has base $2r$, and the two side lengths are $R$. The angle at the center of the circle is $\frac{2\pi}{n}$. The height of the triangle (by Pythagoras) is $\sqrt{R^2-r^2}$ and so the area enclosed in the triangle is $\frac12 2r \sqrt{R^2-r^2}$. The area in the three sectors are: $\frac{\pi}{n}(R-r)^2$, two sets of $\frac12(\frac12 \l \pi - \frac{2\pi}{n}\r)r^2 = \l\frac{1}{2} - \frac{1}{n} \r \frac12 \pi r^2$.

Therefore the remaining area $Y/n$ is $r \sqrt{R^2-r^2} -\frac{\pi}{n}(R-r)^2 - \l\frac{1}{2} - \frac{1}{n} \r \pi r^2$. Multiplying this by $n$ we get the desired result.

For $X$ we can look at the larger sector, which we obtain using the extension of this triangle. This has area $\frac12 \frac{2 \pi}{n} (R+r)^2$. The areas to be removed are the area of the triangle:  $r \sqrt{R^2-r^2}$ and the areas of the two sectors, which have radii $r$ and angles $\pi - \l \frac12 -\frac1{n} \r\pi = \l \frac{1}{2} + \frac{1}{n} \r \pi$. Therefore the area for $X/n$ is $\frac{\pi}{n} (R+r)^2 - r \sqrt{R^2-r^2} - \l \frac{1}{2} + \frac{1}{n} \r \pi r^2$ and so $X$ has area:

\[ X = \pi (R+r)^2 - nr \sqrt{R^2-r^2} - n\pi r^2\l \frac{1}{2} +\frac{1}{n} \r \]

$Z = n\pi r^2$

\begin{align*}
    \frac{(X+Y)}{Z} &= \frac{\pi (R+r)^2 - nr \sqrt{R^2-r^2} - n\pi r^2\l \frac{1}{2} +\frac{1}{n} \r}{n \pi r^2 }\\
    &= \qquad \frac{nr\sqrt{R^{2}-r^{2}}-\pi(R-r)^{2}-n\pi r^{2}\left(\frac{1}{2}-\frac{1}{n}\right)}{n \pi r^2} \\
    &= \frac{4\pi R r - n \pi r^2}{n\pi r^2} \\
    &= \frac{4R-nr}{n r} \\
    &= \frac{4R}{nr} - 1
\end{align*}

Since $\frac{r}{R} = \sin \frac{\pi}{n}$ we have:

\begin{align*}
    \frac{(X+Y)}{Z} &= \frac{4R}{nr} - 1 \\
    &= \frac{4}{n \sin \frac{\pi}n} - 1 \\
    & \to \frac{4}{\pi} - 1
\end{align*}