To find the total surface area of a triangular prism, you need to calculate the areas of the two triangular bases and the three rectangular lateral faces.
Based on your description, let's clarify the dimensions:
- The triangle has a base width of 6 cm and a height of 5 cm.
- The length of the prism (the distance between the two triangular bases) is 12 cm.
Step 1: Calculate the area of the triangular base.
The area \( A \) of a triangle is given by the formula: \[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Using the provided dimensions:
- Base = 6 cm
- Height = 5 cm
\[ A = \frac{1}{2} \times 6 \times 5 = 15 , \text{cm}^2 \]
Since there are two triangular bases, the total area for the triangular bases: \[ \text{Total area of triangles} = 2 \times A = 2 \times 15 = 30 , \text{cm}^2 \]
Step 2: Calculate the area of the rectangular lateral faces.
The dimensions of the rectangular faces depend on the sides of the triangular base and the length of the prism. The lateral faces will correspond to the three sides of the triangle and the length (12 cm).
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For the side with height (5 cm): \[ \text{Area} = \text{height} \times \text{length} = 5 \times 12 = 60 , \text{cm}^2 \]
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For the base (6 cm): \[ \text{Area} = \text{width} \times \text{length} = 6 \times 12 = 72 , \text{cm}^2 \]
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For the other side (Hypotenuse): To find the length of the hypotenuse, we can use Pythagoras' theorem since we have a triangle with height (5 cm) and half of the base (3 cm): \[ \text{Hypotenuse} = \sqrt{(3^2 + 5^2)} = \sqrt{(9 + 25)} = \sqrt{34} , \text{cm} \] The area of the face corresponding to this side is: \[ \text{Area} = \text{Hypotenuse} \times \text{length} = \sqrt{34} \times 12 \approx 12(5.83) \approx 69.96 , \text{cm}^2 \]
Step 3: Combine all areas to find total surface area.
Now, let's add everything together: \[ \text{Total Surface Area} = \text{Area of triangles} + \text{Area of rectangles} \] \[ \text{Total Surface Area} = 30 + (60 + 72 + 69.96) \] \[ \text{Total Surface Area} = 30 + 201.96 \approx 231.96 , \text{cm}^2 \]
Final Result:
Thus, the total surface area of the triangular prism is approximately 232 cm². Please check all parameters and recalculate if necessary based on any additional specifications.