geometry

A circle is inscribed in a quarter-circle sector. It is tangent to the arc of the sector and the two perpendicular radii of the sector. What is the ratio of the area of the inscribed circle to the area of the sector?

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\(a=r+r\sqrt{2}\\ a=r(1+\sqrt{2})\)          Radius of the sector

\(A_{sec}=\pi r^2(1+\sqrt{2})^2/4\)

\(A_{insc}=\pi r^2\)

\(\frac{A_{insc}}{A_{sec}}=\frac{4}{(1+\sqrt{2})^2}\)

\(\frac{A_{insc}}{A_{sec}}=\frac{2}{1+\sqrt{2}}=1:2,414\)

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