To find the surface area of a regular hexagonal prism, you need to consider the surface area contributed by the two hexagonal bases and the rectangular lateral faces.
-
Surface Area of the Hexagonal Bases: A regular hexagon can be divided into 6 equilateral triangles. The area \( A \) of one equilateral triangle with side length \( s \) is given by the formula:
\[ A = \frac{\sqrt{3}}{4} s^2 \]
Therefore, the area of the hexagonal base is:
\[ \text{Area of hexagon} = 6 \cdot \frac{\sqrt{3}}{4} s^2 = \frac{3\sqrt{3}}{2} s^2 \]
If \( s = 5 , \text{cm} \):
\[ \text{Area of hexagon} = \frac{3\sqrt{3}}{2} (5)^2 = \frac{3\sqrt{3}}{2} \cdot 25 = \frac{75\sqrt{3}}{2} , \text{cm}^2 \approx 64.95 , \text{cm}^2 \]
Since there are two bases:
\[ \text{Total area of both bases} = 2 \cdot \frac{75\sqrt{3}}{2} = 75\sqrt{3} , \text{cm}^2 \approx 129.9 , \text{cm}^2 \]
-
Surface Area of the Lateral Faces: The lateral surface area is made up of 6 rectangular faces. The area of each rectangular face is \( \text{width} \times \text{height} \) where the width is the side length of the hexagon \( s \) and the height of the prism \( h \).
Given \( h = 4.3 , \text{cm} \) and \( s = 5 , \text{cm} \):
\[ \text{Area of one rectangular face} = s \cdot h = 5 \cdot 4.3 = 21.5 , \text{cm}^2 \]
Therefore, the total area of the lateral faces is:
\[ \text{Lateral surface area} = 6 \cdot 21.5 = 129 , \text{cm}^2 \]
-
Total Surface Area: Finally, we sum the areas of the bases and the lateral faces:
\[ \text{Total Surface Area} = \text{Area of bases} + \text{Lateral surface area} \]
Plugging in the values we calculated:
\[ \text{Total Surface Area} = 75\sqrt{3} + 129 \approx 129.9 + 129 \approx 258.9 , \text{cm}^2 \]
So, the surface area of the regular hexagonal prism is approximately:
\[ \text{Total Surface Area} \approx 258.9 , \text{cm}^2 \]
If you need a more precise answer with simplified calculations using just numerical values without approximation:
\[ \text{Total Surface Area} = 75\sqrt{3} + 129 \approx 258.9 , \text{cm}^2 \text{ or } 259 , \text{cm}^2} \]
You can state the surface area as approximately \( 258.9 , \text{cm}^2 \).