Answer:
7.8 cm
Step-by-step explanation:
The relevant formula is ...
A = 1/2aP . . . . where a is the apothem, and P is the perimeter.
For a regular 12-gon the perimeter is 12 times the side length, so we have ...
P = 12s = 12(4.2 cm) = 50.4 cm
Then the apothem is found from ...
197 cm² = (1/2)a·(50.4 cm)
a = 197/25.2 cm ≈ 7.81 cm
The apothem is about 7.8 cm.
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Additional comment
A regular 12-gon with a side length of 4.2 cm will have an area of about 197.500 cm². The apothem will be about 7.837 cm. The area of a regular n-gon can be calculated from the side length as ...
A = ns²/(4·tan(180°/n))
Similarly, the apothem will be ...
a = s/(2·tan(180°/n))
