A glass trophy is designed in the shape of a regular right pyramid with a square base. If each edge of the square base measures 12 cm and the perpendicular height from the center of the base to the apex is 8 cm, what are its slant height and total surface area?

The correct answer is 10.0 cm, 384 cm².

Step 1: Identify the given dimensions of the square pyramid
The glass trophy is shaped as a regular right pyramid with a square base.
Base edge length ($a$):
$a=12\text{cm}$
Perpendicular height from the center of the square base to the apex ($h$):
$h=8\text{cm}$

Step 2: Calculate the slant height ($l$)
Inside a regular right pyramid with a square base, a right-angled triangle is formed by:
1. The vertical height ($h$) connecting the apex to the center of the base.
2. The segment connecting the center of the square base to the midpoint of one of its sides. Its length is half of the base edge length:
$\frac{a}{2}=\frac{12}{2}=6\text{cm}$
3. The slant height ($l$), which acts as the hypotenuse of this right triangle.

Applying the Pythagorean theorem:
$l^2=h^2+{\left(\frac{a}{2}\right)}^2$
Substitute the given values:
$l^2=8^2+6^2$
$l^2=64+36=100$
$l=\sqrt{100}=10.0\text{cm}$

Step 3: Calculate the total surface area of the trophy
The total surface area ($TSA$) of a regular pyramid is equal to the area of its base plus the area of its four triangular lateral faces.

Area of the square base:
$\text{Base Area}=a^2={12}^2=144{\text{cm}}^2$

Lateral surface area:
The lateral surface area consists of 4 identical triangles, each having base length $a$ and height equal to slant height $l$:
$\text{Lateral Area}=4\times \left(\frac{1}{2}\times a\times l\right)=2\times a\times l$
$\text{Lateral Area}=2\times 12\times 10=240{\text{cm}}^2$

Total surface area:
$TSA=\text{Base Area}+\text{Lateral Area}$
$TSA=144+240=384{\text{cm}}^2$

Therefore, the slant height is 10.0 cm and the total surface area is 384 cm².