4. Given a cuboid with edges $a, b, c$; let's denote it by $A$. Let

$$
B=\left\{X \mid X \in \mathbb{R}^{3}, d(X, A) \leq 1\right\}
$$

represent the geometric locus of points $X$ in space for which there exists at least one point on the cuboid $A$ that is at a distance not greater than 1 from the point $X$. Determine the volume $V_{B}$ in terms of $a, b, c$.
Solution. The geometric locus $B$ of points consists of:

- the cuboid $A$;
- cuboids with height 1 above each of the faces of $A$;
- quarter cylinders with radius 1 and height along each of the edges of $A$;
- eighth spheres with radius 1 and centers at each vertex of $A$ (see diagram).

![](https://cdn.mathpix.com/cropped/2024_06_05_0970dfed9f8278f60f58g-3.jpg?height=399&width=531&top_left_y=1144&top_left_x=941)

Accordingly,

$$
V_{B}=V_{A}+V_{k}+V_{\text {C }}+V_{\mathrm{T}}=a b c+2(a b+b c+c a)+\pi(a+b+c)+\frac{4 \pi}{3} .
$$

## Third Year