The perimeter of the dodecagon is approx 268.7 units.
How to determine the perimeter of a regular polygon
There is the following relationship between the apothema ([tex]a[/tex]) and the side length ([tex]l[/tex]):
[tex]\tan \frac{\alpha}{2} = \frac{a}{\frac{l}{2} }[/tex] (1)
Where [tex]\alpha[/tex] is the internal angle, in degrees.
The internal angle ([tex]\alpha[/tex]), in degrees, and perimeter ([tex]p[/tex]) are equal to the respective formulas:
[tex]\alpha = \frac{360}{n}[/tex] (2)
[tex]p = n\cdot l[/tex] (3)
Where [tex]n[/tex] is the number of sides.
By (1) we have an expression for [tex]l[/tex]:
[tex]l = \frac{2\cdot a}{\tan \frac{\alpha}{2} }[/tex]
(3) in (1):
[tex]l = \frac{2\cdot a}{\tan \frac{180}{n} }[/tex] (1b)
(1b) in (3):
[tex]p = n\cdot \left(\frac{2\cdot a}{\tan \frac{180}{n} } \right)[/tex] (3b)
If we know that [tex]a = 3[/tex] and [tex]n = 12[/tex], then the perimeter of the polygon is:
[tex]p = 12\cdot \left[\frac{2\cdot (3)}{\tan \frac{180}{12} } \right][/tex]
[tex]p \approx 268.7[/tex]
The perimeter of the dodecagon is approx 268.7 units. [tex]\blacksquare[/tex]
To learn more on polygons, we kindly invite to check this verified question: https://brainly.com/question/17756657
