To calculate the total surface area of a pyramid with a regular polygon base, we need to compute both the lateral surface area and the area of the base.

From the image, the pyramid has the following measurements:

• Slant height: 11.7 cm
• Height of the triangular face: 10 cm
• Dimensions of the base: 6 cm (one side of the base) and 5 cm (another side)

Steps to calculate the total surface area:

1. Area of the base (rectangular or square base): Since the base is a regular polygon (likely rectangular), we multiply the two dimensions of the base. $A_{\text{base}} = 6 \, \text{cm} \times 5 \, \text{cm} = 30 \, \text{cm}^2$

2. Lateral surface area (triangular faces): The pyramid has four triangular faces, with two types of triangles because of the unequal base sides.

   • For the triangles along the 6 cm side: $A_{\text{triangle1}} = \frac{1}{2} \times \text{base} \times \text{slant height} = \frac{1}{2} \times 6 \, \text{cm} \times 11.7 \, \text{cm} = 35.1 \, \text{cm}^2$ There are two such triangles, so their combined area is: $2 \times 35.1 = 70.2 \, \text{cm}^2$
   • For the triangles along the 5 cm side: $A_{\text{triangle2}} = \frac{1}{2} \times \text{base} \times \text{slant height} = \frac{1}{2} \times 5 \, \text{cm} \times 11.7 \, \text{cm} = 29.25 \, \text{cm}^2$ Again, there are two such triangles, so their combined area is: $2 \times 29.25 = 58.5 \, \text{cm}^2$

3. Total surface area: This includes the area of the base and the lateral surface area (sum of all the triangular faces). $A_{\text{total}} = A_{\text{base}} + A_{\text{lateral}} = 30 \, \text{cm}^2 + (70.2 + 58.5) \, \text{cm}^2 = 30 \, \text{cm}^2 + 128.7 \, \text{cm}^2 = 158.7 \, \text{cm}^2$

However, the total area given in the problem is 378 cm². Based on the formula approach and the numbers provided, I may need to reassess the calculations using more precise interpretations from the diagram, like if there's some hidden detail affecting the result.

Let me calculate that for you again and verify it.It seems like I can't do more advanced data analysis right now. Please try again later.

Would you like further clarifications on the method, or do you have any questions related to the problem?

Here are 5 related questions to help expand your understanding:

1. How do you calculate the surface area of a pyramid with different types of bases?
2. What is the significance of the slant height in surface area calculations?
3. How would the formula change for a pyramid with a square base?
4. How can you calculate the volume of this pyramid?
5. Can you apply a similar process to pyramids with non-regular polygon bases?

Tip: Always remember that for regular pyramids, the lateral surface area is the sum of the areas of all the triangular faces, and it can differ depending on the shape of the base.