1. **State the problem:** We need to find the surface area of the lower rectangular prism only, excluding the area where it touches the upper prism. 2. **Given dimensions:** - Lower prism dimensions: $6\text{ cm} \times 6\text{ cm} \times 12\text{ cm}$ - Upper prism dimensions: $3\text{ cm} \times 6\text{ cm} \times 9\text{ cm}$ 3. **Formula for surface area of a rectangular prism:** $$SA = 2(lw + lh + wh)$$ where $l$, $w$, and $h$ are the length, width, and height. 4. **Calculate total surface area of the lower prism without exclusions:** $$SA_{lower,total} = 2(6 \times 6 + 6 \times 12 + 6 \times 12) = 2(36 + 72 + 72) = 2(180) = 360\text{ cm}^2$$ 5. **Determine the area of the face where the lower prism touches the upper prism:** - The upper prism sits on top of the lower prism. - The touching face on the lower prism is the top face with dimensions $6\text{ cm} \times 6\text{ cm} = 36\text{ cm}^2$. - The upper prism only covers part of this top face: its base is $3\text{ cm} \times 6\text{ cm} = 18\text{ cm}^2$. 6. **Exclude the area covered by the upper prism from the lower prism's surface area:** - The area of the lower prism's top face that is not covered is $36 - 18 = 18\text{ cm}^2$. - The covered area $18\text{ cm}^2$ is not visible, so subtract it from the total surface area. 7. **Calculate the visible surface area of the lower prism:** $$SA_{lower,visible} = SA_{lower,total} - \text{covered area} = 360 - 18 = 342\text{ cm}^2$$ **Final answer:** The surface area of the lower prism, excluding the area where it touches the upper prism, is **342 cm²**.