5) From a vertex $A$ of a cube, arcs of a circle with center at $A$ and radius equal to the side of the cube are drawn on each of the three faces having a vertex at $A$. What is the fraction of the surface of the cube that is shaded?
(A) $\frac{1}{4}$
(B) $\frac{3}{8}$
(C) $\frac{\pi}{8}$
(D) $\frac{\pi}{6}$
(E) depends on the side of the cube.

![](https://cdn.mathpix.com/cropped/2024_04_17_f693bde306ad8eee0381g-1.jpg?height=254&width=272&top_left_y=138&top_left_x=2497)
5) The answer is $\mathbf{( C )}$. Indicating with $l$ the length of the side of the cube, the total surface area will be $6 l^{2}$. The shaded region is formed by 3 quarters of circumferences, so its area will be $3 \times \frac{\pi l^{2}}{4}$. The ratio will therefore be $\frac{3 \times \frac{\pi l^{2}}{4}}{6 l^{2}}=\frac{3 \pi}{24}=\frac{\pi}{8}$.