We are asked to determine the surface area of pencils that have the shape of a hexagonal prism. To do that we will add the area of the sides and the area of the top and bottom parts.
The areas of the sides are the areas of rectangles of 0.5cm in width and 20 cm in longitude since there are 6 sides we have:
Where:
Substituting the values we get:
![A=6(0.6\operatorname{cm})(20\operatorname{cm})]()
Solving the operations:
![A=72\operatorname{cm}^2]()
Now we add the area of the hexagon of the top and bottom parts. The area of a hexagon is given by:
![A_{\text{hex}}=\frac{3\sqrt[]{3}a^2}{2}](https://img.qammunity.org/2023/formulas/mathematics/college/tvoczytfh7s9rfzs51di3qj0wsrym5iwhc.png)
Where "a" is the length of the side.
Since there are two areas we multiply the area of the hexagon by 2, therefore, substituting we get:
![A_{\text{hex}}=2*\frac{3\sqrt[]{3}(0.5\operatorname{cm})^2}{2}]()
Solving the operations:
![A_{\text{hex}}=1.3\operatorname{cm}^2]()
Now, the total surface area is the sum of both areas:
Substituting the values:
![\begin{gathered} A_{\text{sur}}=72\operatorname{cm}^2+1.3cm^2 \\ A_{\text{sur}}=73.3\operatorname{cm}^2 \end{gathered}]()
Therefore, the surface area of the pencil is 73.3 square centimeters.