To calculate the surface area of the triangular prism, we need to find the area of the triangular base and the area of the three rectangular faces.
1. Area of the triangular base:
The formula for the area of a triangle is:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
For the triangular prism, the base of the triangle measures 10 cm and the height from the top vertex to the base is 8 cm:
\[ \text{Area} = \frac{1}{2} \times 10 , \text{cm} \times 8 , \text{cm} = 40 , \text{cm}^2 \]
Since there are two triangular bases, we multiply this area by 2:
\[ \text{Total area of triangular bases} = 2 \times 40 , \text{cm}^2 = 80 , \text{cm}^2 \]
2. Area of the rectangular faces:
- The left rectangular face is determined by the height of the prism and the width of the triangular base. Its area is:
\[ \text{Area} = \text{base} \times \text{height} = 10 , \text{cm} \times 6 , \text{cm} = 60 , \text{cm}^2 \]
- The right rectangular face is the same as the left rectangular face, so its area is also:
\[ \text{Area} = 10 , \text{cm} \times 6 , \text{cm} = 60 , \text{cm}^2 \]
- The third rectangular face connects the two bases and is determined by the height of the prism (6 cm) and the length of the prism which is the height of the triangular face (8 cm):
\[ \text{Area} = 8 , \text{cm} \times 6 , \text{cm} = 48 , \text{cm}^2 \]
3. Calculate the total surface area:
Now, we sum the areas of the triangular bases and all the rectangular faces:
\[ \text{Total surface area} = \text{Area of triangular bases} + \text{Area of left rectangle} + \text{Area of right rectangle} + \text{Area of connecting rectangle} \]
\[ \text{Total surface area} = 80 , \text{cm}^2 + 60 , \text{cm}^2 + 60 , \text{cm}^2 + 48 , \text{cm}^2 \]
\[ \text{Total surface area} = 80 + 60 + 60 + 48 = 248 , \text{cm}^2 \]
It looks like there's a discrepancy in the surface area calculated (248 cm²) compared to the provided responses. Let's check for any possible error in the areas calculated.
Make sure that the rectangular face calculations correspond to the dimensions provided for the prism, rechecking areas calculated above leads us compactly back to the surface area of 248 cm², reconfirming the above stated dimensions and values.
However, this matches none of the provided choices, implying a misunderstanding in the options available.
Double-checking calculated and concluding part leads to 248 square centimeters being out of options.
The closest possible answer appears to be 260 square centimeters assuming a minor miscalculation in the context or dimension interpretation widely used.