Any smart ideas on finding the area of this shaded region?

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Don’t let the simplicity of this diagram fool you. I have been wondering about this for quite some time, but I can’t think of an easy/smart way of finding it.

Any ideas?

For reference, the Area is:

\bbox[10pt, border:2pt solid grey]{90−18.75\pi−25\cdot \arctan\left(\frac 12\right)}


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We observe that \triangle PRT can be partitioned into five congruent sub-triangles. Therefore, the entire shaded region has area given by …
3 u + |\text{region}\; PAT| &= 3u + |\square OAPT| – |\text{sector}\;OAT| \\[6pt]
&= 3u + \frac{3}{5}\,|\triangle PRT| – |\text{sector}\;OAT| \\[6pt]
&= 3\cdot\frac{1}{4} r^2 \left( 4 – \pi \right) \;+\; \frac{3}{5}\cdot r^2 \;-\; \frac{1}{2}r^2\cdot 2\theta

Since \theta = \operatorname{atan}\frac{1}{2}, this becomes

r^2\left(\; \frac{18}{5} – \frac{3}{4}\pi – \operatorname{atan}\frac{1}{2} \;\right) \qquad\stackrel{r=5}{\to}\qquad
90 – \frac{75}{4}\pi – 25\;\operatorname{atan}\frac{1}{2}

Source : Link , Question Author : The Artist , Answer Author : Blue

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