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The perimeter of a regular pentagon is 30 cm.

1. What is the area of the pentagon?
2. What is the area of the circle that can be circumscribed about the pentagon?

Answer :

Final answer:

To find the area of the regular pentagon with a perimeter of 30 cm, divide the perimeter by 5 to get the side length, then use the regular polygon area formula. For the circumscribed circle, calculate the radius using the pentagon side length and then apply the area formula for a circle.Substituting s with 6 cm gives us the area of the pentagon.

Explanation:

The perimeter of a regular pentagon is 30 cm. To find the length of one side of the pentagon, we divide the perimeter by 5 (since a pentagon has five sides). Thus, each side of the pentagon is 6 cm. To calculate the area of the pentagon, we can use the formula for the area of a regular polygon, A = \(\frac{1}{4}\sqrt{5(5+2\sqrt{5})} \times s^2\), where s is the side length. Substituting s with 6 cm gives us the area of the pentagon.

To find the area of the circumscribed circle, or the circumcircle, we first need to determine the radius of the circle. The radius (r) of the circumcircle of a regular pentagon with side length s is given by r = \(\frac{s}{2\sin(\frac{\pi}{5})}\). Once we have the radius, we can use the formula for the area of a circle, A = \(\pi r^2\), to find the area of the circumcircle. Inserting the calculated radius into this formula will provide the desired area.

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