A circle is inscribed in a square with a side length of 200. If a point in the square is chosen at random, what is the probability that the point is inside the circle? Give your answer as a percentage rounded to the nearest tenth.)

The point chosen is either inside the circle, or inside the square but outside the circle. If the probability of choosing a given position is uniform across the entire square, the probability of landing inside the circle would be equal to the ratio between the area of the circle and the area of the square:

[tex]\begin{aligned} P(\text{inside circle}) &= \frac{(\text{area of inscribed circle})}{(\text{area of square})}\end{aligned}[/tex].

Refer to the diagram attached. Let [tex]l = 200[/tex] denote the length of the side of the square. The radius of the inscribed circle would be [tex]r = (l / 2)[/tex].

Area of the square: [tex]l^{2}[/tex].
Area of the inscribed circle: [tex]\pi\, r^{2} = (\pi)\, (l/2)^{2}[/tex].

Hence, the probability that the chosen point is inside the circle would be:

[tex]\begin{aligned} & P(\text{inside circle}) \\=\; & \frac{(\text{area of inscribed circle})}{(\text{area of square})} \\ =\; & \frac{(\pi)\, (l/2)^{2}}{l^{2}} \\ =\; & \frac{\pi}{4} \\ \approx\; & 78.5\%\end{aligned}[/tex].