1. **Problem statement:** Calculate the surface area of a cube placed on top of a right triangular prism. The prism has a rectangular base of 15 cm by 6 cm and a right triangle on top with legs 12 cm and 9 cm. The cube has side length 3 cm and sits on the triangular face. 2. **Step 1: Calculate the surface area of the triangular prism alone.** - The prism has two rectangular faces (15 cm by 6 cm and 15 cm by the hypotenuse of the triangle), two triangular faces, and one rectangular base. - Hypotenuse of the triangle: $$\sqrt{12^2 + 9^2} = \sqrt{144 + 81} = \sqrt{225} = 15 \text{ cm}$$ - Rectangular faces: - Bottom base: $$15 \times 6 = 90 \text{ cm}^2$$ - Side rectangle along height: $$15 \times 6 = 90 \text{ cm}^2$$ - Side rectangle along hypotenuse: $$15 \times 15 = 225 \text{ cm}^2$$ - Triangular faces (2 identical): $$\frac{1}{2} \times 12 \times 9 = 54 \text{ cm}^2$$ each, total $$108 \text{ cm}^2$$ - Total surface area of prism without cube: $$90 + 90 + 225 + 108 = 513 \text{ cm}^2$$ 3. **Step 2: Calculate the surface area of the cube.** - Cube side length: 3 cm - Surface area of cube: $$6 \times 3^2 = 6 \times 9 = 54 \text{ cm}^2$$ 4. **Step 3: Adjust for the area of the cube face in contact with the prism.** - The cube rests on the triangular face, so one face of the cube (area $$3 \times 3 = 9 \text{ cm}^2$$) covers part of the prism's triangular face. - Subtract this area from the prism's surface area to avoid double counting. 5. **Step 4: Calculate total surface area of the combined solid.** - Total surface area = Prism surface area - area covered by cube + cube surface area - $$= 513 - 9 + 54 = 558 \text{ cm}^2$$ **Final answer:** The total surface area of the cube on the triangular prism is $$\boxed{558 \text{ cm}^2}$$.