1. **Problem:** Given the surface area of a cube is 54 cm², find its volume. 2. **Formula for surface area of a cube:** $$\text{Surface Area} = 6a^2$$ where $a$ is the side length of the cube. 3. **Calculate side length:** $$6a^2 = 54 \implies a^2 = \frac{54}{6} = 9 \implies a = \sqrt{9} = 3 \text{ cm}$$ 4. **Formula for volume of a cube:** $$\text{Volume} = a^3$$ 5. **Calculate volume:** $$3^3 = 27 \text{ cm}^3$$ --- 6. **Problem:** Given the cubic diagonal is $4\sqrt{3}$, find the volume of the cube. 7. **Formula for cubic diagonal:** $$d = a\sqrt{3}$$ where $d$ is the diagonal length. 8. **Calculate side length:** $$a\sqrt{3} = 4\sqrt{3} \implies a = 4$$ 9. **Calculate volume:** $$a^3 = 4^3 = 64 \text{ cm}^3$$ **Final answers:** - Volume from surface area: $27$ cm³ - Volume from cubic diagonal: $64$ cm³