1. **State the problem:** Calculate the surface area of a triangular prism with triangular base sides 4 in, 5 in, and 6 in, and height (length) 3 in. 2. **Formula and important rules:** The surface area $SA$ of a prism is given by: $$SA = 2 \times \text{Area of base} + \text{Perimeter of base} \times \text{height}$$ 3. **Given data:** - Area of base $= 6$ in$^2$ - Height (length) of prism $= 3$ in - Sides of triangular base: 4 in, 5 in, 6 in 4. **Calculate the perimeter of the base:** $$P = 4 + 5 + 6 = 15 \text{ in}$$ 5. **Calculate the lateral surface area:** $$\text{Lateral area} = P \times h = 15 \times 3 = 45 \text{ in}^2$$ 6. **Calculate total surface area:** $$SA = 2 \times 6 + 45 = 12 + 45 = 57 \text{ in}^2$$ **Final answer:** The surface area of the prism is $57$ in$^2$.