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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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