Determine the total capacity of a three-dimensional container shaped as a right prism, given that its base is a regular six-sided polygon with edge length 8 cm and its vertical height measures 12 cm.

The correct answer is $1152\sqrt{3}$ cm³.

Step 1: Understand the Given Dimensions
We are given a three-dimensional container shaped as a right prism.
• Base shape: A regular hexagon (a 6-sided regular polygon)
• Edge length of the base ($a$) = 8 cm
• Height of the prism ($h$) = 12 cm

Step 2: Formula for the Capacity (Volume) of a Prism
The total capacity or volume ($V$) of any right prism is calculated using the formula:

$V=\text{Base Area}\times \text{Height}$

Step 3: Calculate the Base Area (Area of a Regular Hexagon)
A regular hexagon with side length $a$ can be divided into 6 congruent equilateral triangles.
The area of one equilateral triangle of side $a$ is given by:

$\text{Area of one equilateral triangle}=\frac{\sqrt{3}}{4}{a}^2$

Therefore, the area of the regular hexagonal base ($B$) is 6 times the area of one equilateral triangle:

$B=6\times \frac{\sqrt{3}}{4}{a}^2=\frac{3\sqrt{3}}{2}{a}^2$

Substitute $a=8$ cm into the base area formula:

$B=\frac{3\sqrt{3}}{2}\times 8^2$

$B=\frac{3\sqrt{3}}{2}\times 64$

$B=3\sqrt{3}\times 32=96\sqrt{3}\text{ cm}{2}^{}$

Step 4: Calculate the Total Capacity (Volume) of the Prism
Now multiply the base area ($B$) by the height ($h=12$ cm):

$V=B\times h$

$V=96\sqrt{3}\times 12$

$V=1152\sqrt{3}\text{ cm}{3}^{}$

Thus, the total capacity of the container is $1152\sqrt{3}$ cm³.