To find the surface area of a triangular prism, we need to calculate the areas of the two triangular bases and the three rectangular lateral surfaces.
Given:
- Height of prism \( h = 9 \) inches (which is the height of the triangle)
- Base of triangle \( b = 17 \) inches
- Sides of the triangle \( s_1 = 10 \) inches, \( s_2 = 12 \) inches, \( s_3 = 17 \) inches
- Length of the prism \( l = 10 \) inches
Step 1: Area of the triangular base
To find the area of the triangle, we can use the formula: \[ A = \frac{1}{2} \times base \times height \] Using the height and the base of the triangle: \[ A = \frac{1}{2} \times 17 \times 9 \] \[ A = \frac{1}{2} \times 153 \] \[ A = 76.5 , \text{square inches} \]
Step 2: Area of the two triangular bases
Since there are two triangular bases, the combined area is: \[ A_{total,triangles} = 2 \times A = 2 \times 76.5 = 153 , \text{square inches} \]
Step 3: Area of the rectangular lateral surfaces
Now, we calculate the area of the three rectangles (lateral surfaces) formed by each side of the triangle and the length of the prism.
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For side \( s_1 = 10 \): \[ A_1 = s_1 \times l = 10 \times 10 = 100 , \text{square inches} \]
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For side \( s_2 = 12 \): \[ A_2 = s_2 \times l = 12 \times 10 = 120 , \text{square inches} \]
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For side \( s_3 = 17 \): \[ A_3 = s_3 \times l = 17 \times 10 = 170 , \text{square inches} \]
Step 4: Total area of the lateral surfaces
\[ A_{total,rectangles} = A_1 + A_2 + A_3 = 100 + 120 + 170 = 390 , \text{square inches} \]
Step 5: Total surface area of the prism
Now, we add the area of the triangular bases and the lateral surfaces together: \[ S.A. = A_{total,triangles} + A_{total,rectangles} = 153 + 390 = 543 , \text{square inches} \]
Final answer:
\[ S.A. = 543 , \text{in}^2 \]