To find the perimeter of a square, you can use the formula:
\[ \text{Perimeter} = 4 \times \text{side length} \]
In this case, the side length is \( \sqrt{212} \).
So we calculate:
\[ \text{Perimeter} = 4 \times \sqrt{212} \]
Next, we can simplify \( \sqrt{212} \):
- \( 212 = 4 \times 53 \)
- Therefore, \( \sqrt{212} = \sqrt{4 \times 53} = \sqrt{4} \times \sqrt{53} = 2\sqrt{53} \)
Now substituting back in for the perimeter:
\[ \text{Perimeter} = 4 \times 2\sqrt{53} = 8\sqrt{53} \]
Now, let's look at the options provided:
- \( 16\sqrt{3} \)
- \( 8\sqrt{48} \)
- \( 32\sqrt{3} \)
- \( 8\sqrt{12} \)
None of the answers match \( 8\sqrt{53} \) exactly. However, if we simplify \( 8\sqrt{48} \):
\[ 8\sqrt{48} = 8\sqrt{16 \times 3} = 8 \times 4\sqrt{3} = 32\sqrt{3} \]
Also, simplify \( 8\sqrt{12} \):
\[ 8\sqrt{12} = 8\sqrt{4 \times 3} = 8 \times 2\sqrt{3} = 16\sqrt{3} \]
It appears that no option correctly represents the perimeter calculated as \( 8\sqrt{53} \). Please check if \( 8\sqrt{53} \) or a variant of it is an option.
Otherwise, based on what we have calculated, the perimeter in simplest radical form is \( 8\sqrt{53} \). If the options don't include that, then there may have been an error in the options provided.