Answer:
[tex] \boxed{A ≈ 58.1 \: cm^{2}} [/tex]
Step-by-step explanation:
When given a regular polygon.
s = a × 2 tan (π / n [radians]).
s = a × 2 tan (180 / n [degrees])
A = (n × s × a) / 2
Where s is the side length, n is the number of sides, a is the apothem length, and A is the area.
You may also see A = p × a.
Where p is the perimeter, the perimeter is the same as n × s.
The apothem is also known as the inradius.
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Given a regular pentagon which has 5 sides, and an apothem of 4 cm.
s = 4 × 2 tan (180 / 5) = 4 × 2 × (0.726542528) = 8 × (0.726542528) = 5.81234022404.. ≈ 5.8 cm.
Now that we have sufficient information, we can now solve for the area.
A = (n × s × a) / 2
A = (5 × 5.81234022404.. cm × 4) / 2
A = (20 × 5.81234022404.. cm) / 2
A = (116.236804481.. cm²) / 2
A = 58.1234022404.. cm²
A ≈ 58.1 cm²
