1. **Problem Statement:** Calculate the surface area of the given shape with sides 6 cm, 12 cm, and 2 cm, where the triangle's hypotenuse is calculated using the Pythagorean theorem. 2. **Formula Used:** For a right triangle, the Pythagorean theorem states: $$a^2 + b^2 = c^2$$ where $a$ and $b$ are the legs and $c$ is the hypotenuse. 3. **Calculate the hypotenuse:** $$6^2 + 2^2 = 36 + 4 = 40$$ $$c = \sqrt{40} = 2\sqrt{10} \text{ cm}$$ 4. **Surface Area Calculation:** Assuming the shape is a prism with the triangular base and length 12 cm, the surface area includes: - Two triangular bases - Three rectangular sides 5. **Area of one triangular base:** $$\frac{1}{2} \times 6 \times 2 = 6 \text{ cm}^2$$ 6. **Area of rectangular sides:** - Side 1: $6 \times 12 = 72$ cm$^2$ - Side 2: $2 \times 12 = 24$ cm$^2$ - Side 3 (hypotenuse side): $2\sqrt{10} \times 12 = 24\sqrt{10}$ cm$^2$ 7. **Total surface area:** $$2 \times 6 + 72 + 24 + 24\sqrt{10} = 12 + 96 + 24\sqrt{10} = 108 + 24\sqrt{10} \text{ cm}^2$$ **Final answer:** $$\boxed{108 + 24\sqrt{10} \text{ cm}^2}$$