Find the apothem of a regular 12-gon (12-sided polygon) if each side is 4.2 cm, and has an area of 197 sq. cm.

  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))

Start fraction 5 over 6 end fraction and 0.833...