The circle is inscribed in a square with sides 8 cm. What is the area of the shaded part in square centimeters? Express your answer in terms of $\pi$

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A nice geometry problem, It's just for sharing a new ideas, thanks:)


Notice, radius of inscribed circle $$=\frac{\text{side of square}}{2}=\frac{8}{2}=4\ cm$$ Area of shaded part $$=\text{(area of semi-circle with radius}\ 4 \ cm)-\text{(area of isosceles triangle)}$$ $$=\frac{1}{2}\pi (4)^2-\frac{1}{2}(2)(8)$$$$=8\pi-8=8(\pi-1)\approx 17.13274123 \ cm^2$$ $$\bbox[5px, border:2px solid #C0A000]{\color{red}{\text{Area of shaded portion}=8(\pi-1)\approx 17.13274123\ cm^2}}$$

  • $\begingroup$ Assuming that the triangles kiss at the midpoint of the line, sure. $\endgroup$ – Race Bannon Aug 5 '15 at 22:23
  • $\begingroup$ Isn´t $r=8$ ?... $\endgroup$ – callculus Aug 5 '15 at 22:23
  • $\begingroup$ Kindly, notice $\text{side of square}=\text{diameter of circle}=8\ cm$ $$r=4\ cm$$ $\endgroup$ – Harish Chandra Rajpoot Aug 5 '15 at 22:25
  • $\begingroup$ I see after I have read the text. But from the picture it looks like that $r=8$ On the other hand it can be interpretated as $r=4$ $\endgroup$ – callculus Aug 5 '15 at 22:28
  • $\begingroup$ Nice solution :) $\endgroup$ – Oiue Aug 5 '15 at 22:49

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