To find the surface area of the given triangular prism, we need to calculate the area of each face and then sum them up. Let's identify and calculate the areas for each of the faces.

Dimensions given:

• The base of the triangular face: 6 cm
• The height of the triangular face: 9 cm
• The length of the prism: 19 cm
• The height of the rectangular face that is the side of the prism: 18 cm

1. Area of the two triangular faces:

The area of one triangular face can be calculated using the formula: $\text{Area of triangle} = \frac{1}{2} \times \text{base} \times \text{height}$ $\text{Area of triangle} = \frac{1}{2} \times 6 \, \text{cm} \times 9 \, \text{cm} = 27 \, \text{cm}^2$ Since there are two identical triangular faces: $\text{Total area of both triangles} = 2 \times 27 \, \text{cm}^2 = 54 \, \text{cm}^2$

2. Area of the three rectangular faces:

• Face 1 (base × length): $\text{Area of Face 1} = 6 \, \text{cm} \times 19 \, \text{cm} = 114 \, \text{cm}^2$

• Face 2 (side height × length): $\text{Area of Face 2} = 9 \, \text{cm} \times 19 \, \text{cm} = 171 \, \text{cm}^2$

• Face 3 (other side height × length): $\text{Area of Face 3} = 18 \, \text{cm} \times 19 \, \text{cm} = 342 \, \text{cm}^2$

3. Total Surface Area:

Now, sum the areas of all the faces: $\text{Total Surface Area} = 54 \, \text{cm}^2 + 114 \, \text{cm}^2 + 171 \, \text{cm}^2 + 342 \, \text{cm}^2$ $\text{Total Surface Area} = 681 \, \text{cm}^2$

Thus, the surface area of the triangular prism is 681 cm².

Would you like to explore further details or have any questions? Here are a few related questions you might find interesting:

1. How do we find the volume of the same triangular prism?
2. What happens to the surface area if the base of the triangle doubles?
3. Can you find the length of the hypotenuse of the triangular face using the given dimensions?
4. What is the significance of knowing the surface area in practical applications?
5. How would the surface area change if the prism’s length was increased by 10 cm?

Tip: Always double-check each calculation step, especially when dealing with multiple faces in geometry, to ensure accuracy.