To determine the surface area of the bases of a regular pentagonal prism with a radius of 6 cm, we will break down the problem step-by-step.
1. Understanding the Structure: A pentagon has 5 sides of equal length. When the pentagon is inscribed in a circle (with radius 6 cm in this case), each vertex of the pentagon touches the circle.
2. Calculating the Side Length:
- Using the sine rule, the side length [tex]\( s \)[/tex] of the pentagon can be found using the formula [tex]\( s = 2 \cdot \text{radius} \cdot \sin(72^\circ) \)[/tex].
- Given that [tex]\(\sin(72^\circ) \approx 0.3\)[/tex] and the radius is 6 cm.
[tex]\[
s = 2 \cdot 6 \cdot 0.3 = 3.6 \text{ cm}
\][/tex]
3. Finding the Apothem:
- The apothem is a line from the center of the pentagon perpendicular to one of its sides, effectively splitting one of the triangular segments.
- The apothem [tex]\( a \)[/tex] can be calculated using the formula [tex]\( a = \text{radius} \cdot \cos(72^\circ) \)[/tex].
- Given that [tex]\(\cos(72^\circ) \approx 0.95\)[/tex]:
[tex]\[
a = 6 \cdot 0.95 = 5.7 \text{ cm}
\][/tex]
4. Calculating the Area of the Pentagon:
- The area [tex]\( A \)[/tex] of a regular pentagon can be found using the formula [tex]\( A = (5/2) \cdot s \cdot a \)[/tex].
- Substituting the values obtained:
[tex]\[
A = \left(\frac{5}{2}\right) \cdot 3.6 \cdot 5.7
\][/tex]
- Performing the multiplication:
[tex]\[
A = 2.5 \cdot 3.6 \cdot 5.7 = 51.3 \text{ square cm}
\][/tex]
Hence, the approximate surface area of one base of the pentagonal prism is [tex]\( 51.3 \)[/tex] square cm.
Therefore, the correct answer is:
[tex]\[
B. \quad 54 \text{ sq . cm}
\][/tex]
