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    <p><img src="https://i.stack.imgur.com/zMsoz.png" width="400" ></p> <pre><code>Phi= ArcTan[ Sqrt[4 * R^2 - d^2] /d ] </code></pre> <p>HTH!</p> <p><b>Edit</b></p> <p>For two different radii: </p> <p><img src="https://i.stack.imgur.com/1VxWW.png" width="400" ></p> <p>Simplifying a little: </p> <pre><code>Phi= ArcTan[Sqrt[-d^4 -(R1^2 - R2^2)^2 + 2*d^2*(R1^2 + R2^2)]/(d^2 +R1^2 -R2^2)] </code></pre> <p><strong>Edit</strong> </p> <p>If you want the angle viewed from the other circle center, just exchange R1 by R2 in the last equation. </p> <p>Here is a sample implementation in Mathematica: </p> <pre><code>f[center1_, d_, R1_, R2_] := Module[{Phi, Theta}, Phi= ArcTan[Sqrt[-d^4-(R1^2-R2^2)^2 + 2*d^2*(R1^2 + R2^2)]/(d^2 +R1^2 -R2^2)] Theta=ArcTan[Sqrt[-d^4-(R1^2-R2^2)^2 + 2*d^2*(R1^2 + R2^2)]/(d^2 -R1^2 +R2^2)] {Circle[{center1, 0}, R1, {2 Pi - Phi, Phi}], Circle[{d, 0}, R2, {Pi - Theta, -Pi + Theta}]} ]; Graphics[f[0, 1.5, 1, 1]] </code></pre> <p><img src="https://i.stack.imgur.com/EyDsY.png" alt="alt text"></p> <pre><code>Graphics[f[0, 1.5, 1, 3/4]] </code></pre> <p><img src="https://i.stack.imgur.com/dZ7pQ.png" alt="alt text"> </p> <p>And...</p> <pre><code>ImageMultiply[ Binarize@FillingTransform[#], ImageResize[Import@ "http://i305.photobucket.com/albums/nn235/greeneyedgirlox/blondebabybunny.jpg", ImageDimensions@#]] &amp;@ Rasterize@Graphics[f[0, 1.5, 1, 1], Background -&gt; Black] </code></pre> <p><img src="https://i.stack.imgur.com/CT0IE.png" alt="alt text"></p> <p>:) </p>
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