Geometry

Inscribed and Circumscribed Figures

Incircle of Squares

         

In the figure above, the black circle is inscribed in square ABCD,\square ABCD, and the white square is inscribed in the circle. If the length of AB\overline{AB} is AB=8,\lvert\overline{AB}\rvert=8, what is the area of the shaded region?

In the figure above, circle OO of radius 88 is inscribed in square ABCD.ABCD. What is the area of the shaded region?

Above illustration is a square circumscribed about a circle. If CD=12,\overline{CD} = 12, What is the area of the circle?

In the above diagram, circle PP with radius 88 is inscribed in square ABCD.ABCD. If a point is picked at random from the interior of the square, what is the probability that the point will lie in the shaded regions?

In the figure above, if circle OO with radius of 55 is inscribed in square ABCDABCD in such a way that each side of the square is tangent to the circle, which of the following statements must be true?

I. AB×CD<25π\lvert{\overline{AB}}\rvert \times \lvert{\overline{CD}}\rvert < 25\pi
II. Area of ABCD=100\text{Area of } \square ABCD = 100
III. 5<2×CDπ\displaystyle 5 < \frac{2 \times \lvert{\overline{CD}}\rvert}{\pi}

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