To calculate the surface area of a triangular prism, you need to know the dimensions of its triangular base and its height (length of the prism). Given that you've provided "5 cm" and "4 cm," I assume these are the lengths of the two sides of the triangular base. However, I also need to know the height (length) of the prism.
Let's denote:
- \( a = 5 , \text{cm} \) (one side of the triangular base),
- \( b = 4 , \text{cm} \) (another side of the triangular base),
- \( c \) = the length of the third side (which we can find using the Pythagorean theorem if the triangle is right-angled or find its actual length from further information),
- \( h \) = the height (length) of the prism.
Step 1: Calculate the area of the triangular base
Assuming this is a right triangle (for simplicity), the area \( A \) of the triangle can be calculated as: \[ A = \frac{1}{2} \times \text{base} \times \text{height} \] If we assume that 5 cm is the base and 4 cm is the height (or vice versa), the triangle's area would be: \[ A = \frac{1}{2} \times 5 , \text{cm} \times 4 , \text{cm} = 10 , \text{cm}^2 \]
Step 2: Calculate the lateral surface area
The lateral surface area of the prism can be calculated as the perimeter of the triangular base multiplied by the height of the prism.
First, we need to compute the perimeter \( P \) of the triangle. Assuming we've already calculated or known the lengths of the triangle's sides (let's denote the third side as \( c \)). If it is a right triangle, and \( c \) is the hypotenuse, then: \[ c = \sqrt{(5^2 + 4^2)} = \sqrt{41} \approx 6.4 , \text{cm} \] Thus, the perimeter would be: \[ P = a + b + c = 5 + 4 + \sqrt{41} \approx 5 + 4 + 6.4 = 15.4 , \text{cm} \]
The lateral surface area \( L \) is given by: \[ L = P \times h \]
Step 3: Calculate total surface area
The total surface area \( S \) of the prism is given by: \[ S = 2A + L \] So: \[ S = 2 \times 10 + (5 + 4 + \sqrt{41}) \times h \] This leads to: \[ S = 20 + (9 + \sqrt{41}) \times h \]
Conclusion
Without knowing the height \( h \) of the prism, we cannot find a precise numerical value for the surface area of the triangular prism. Please provide the height or specify other dimensions for a complete solution.