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Inscribed and Circumscribed Figures

Learn about packing shapes neatly into others, and the resulting geometric properties.

Incircle of Squares

         

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

In the figure above, circle \(O\) of radius \(8\) is inscribed in square \(ABCD.\) What is the area of the shaded region?

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

In the above diagram, circle \(P\) with radius \(8\) is inscribed in square \(ABCD.\) 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 \(O\) with radius of \(5\) is inscribed in square \(ABCD\) in such a way that each side of the square is tangent to the circle, which of the following statements must be true?

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

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