To find the surface area of the triangular prism, we need to calculate the area of all the faces.

The surface area of a triangular prism is given by the formula:

$\text{Surface Area} = 2 \times \text{Base Area of Triangle} + \text{Perimeter of Triangle} \times \text{Length of Prism}$

Step-by-step solution:

1. Find the area of the triangular base: The triangle in the diagram is a right triangle, with base $12 \, \text{cm}$ and height $8 \, \text{cm}$. The area of a triangle is given by:

   $\text{Area of Triangle} = \frac{1}{2} \times \text{Base} \times \text{Height}$ $\text{Area of Triangle} = \frac{1}{2} \times 12 \, \text{cm} \times 8 \, \text{cm} = 48 \, \text{cm}^2$

2. Find the perimeter of the triangular base: The perimeter of the triangle includes the base, height, and hypotenuse. We already know the base and height, so we need to calculate the hypotenuse using the Pythagorean theorem:

   $\text{Hypotenuse} = \sqrt{\text{Base}^2 + \text{Height}^2} = \sqrt{12^2 + 8^2} = \sqrt{144 + 64} = \sqrt{208} \approx 14.42 \, \text{cm}$

   Now, the perimeter of the triangle is:

   $\text{Perimeter of Triangle} = 12 \, \text{cm} + 8 \, \text{cm} + 14.42 \, \text{cm} = 34.42 \, \text{cm}$

3. Calculate the surface area: The length of the prism is given as $20 \, \text{cm}$. Using the surface area formula:

   $\text{Surface Area} = 2 \times \text{Area of Triangle} + \text{Perimeter of Triangle} \times \text{Length of Prism}$ $\text{Surface Area} = 2 \times 48 \, \text{cm}^2 + 34.42 \, \text{cm} \times 20 \, \text{cm}$ $\text{Surface Area} = 96 \, \text{cm}^2 + 688.4 \, \text{cm}^2$ $\text{Surface Area} = 784.4 \, \text{cm}^2$

Thus, the surface area of the triangular prism is approximately 784.4 cm².

Would you like further details or have any questions?

Here are five related questions:

1. How do you calculate the surface area of a rectangular prism?
2. What changes in the surface area formula for non-right triangular prisms?
3. How does the volume of the triangular prism compare to its surface area?
4. Can the Pythagorean theorem be used for other shapes besides right triangles?
5. How does increasing the length of the prism affect the total surface area?

Tip: Always check if the triangle is a right triangle before applying the Pythagorean theorem!