The image shows a cuboid with the dimensions $8 \, \text{cm}$, $6 \, \text{cm}$, and $12 \, \text{cm}$, and several questions that are listed next to it. The questions ask to calculate:

a. Perpendicular diagonal between points.

b. Longest diagonal of the cuboid.

c. Surface area.

d. Volume.

Let me go ahead and solve these problems.

a. Perpendicular Diagonal (Diagonal of a Face)

Assume the question asks for the diagonal across a face of the cuboid, such as the diagonal between the sides 6 cm and 8 cm. The diagonal of a rectangle can be calculated using the Pythagorean theorem: $d = \sqrt{l^2 + w^2}$ Where $l = 8 \, \text{cm}$ and $w = 6 \, \text{cm}$.

b. Longest Diagonal (Body Diagonal)

For the longest diagonal, which is the body diagonal of the cuboid, we use the 3D Pythagorean theorem: $d = \sqrt{l^2 + w^2 + h^2}$ Where $l = 12 \, \text{cm}$, $w = 8 \, \text{cm}$, and $h = 6 \, \text{cm}$.

c. Surface Area

The surface area of the cuboid is the sum of the areas of all six faces: $A = 2(lw + lh + wh)$ Where $l = 12 \, \text{cm}$, $w = 8 \, \text{cm}$,