Calculate the volume of the cube and hence show that the volume of one pyramid, rounded to the nearest 100, is approximately 1 300 cm³.

The dimensions of the cube are given as length = 20 cm, breadth = 20 cm, and height = 20 cm.
Using the formula: Volume of Cube = length $\times$ breadth $\times$ height

$$\text{Volume of Cube} = 20 \text{ cm} \times 20 \text{ cm} \times 20 \text{ cm}$$ $$\text{Volume of Cube} = 8000 \text{ cm}^3$$

The problem states that the solid cube is cut into six square-based pyramids. Therefore, the volume of one pyramid is the total volume of the cube divided by 6.

$$\text{Volume of one pyramid} = \frac{\text{Volume of Cube}}{6}$$ $$\text{Volume of one pyramid} = \frac{8000 \text{ cm}^3}{6}$$ $$\text{Volume of one pyramid} = 1333.333... \text{ cm}^3$$

Rounding $1333.333... \text{ cm}^3$ to the nearest 100:
The hundreds digit is 3. The digit to its right is 3, which is less than 5, so we round down (keep the hundreds digit as it is and change subsequent digits to zero).

$$\text{Volume of one pyramid (rounded)} \approx 1300 \text{ cm}^3$$

This shows that the volume of one pyramid, rounded to the nearest 100, is approximately $1300 \text{ cm}^3$.

The volume of the cube is $\boxed{\text{8000 cm}^3}$ and the volume of one pyramid, rounded to the nearest 100, is approximately $\boxed{\text{1300 cm}^3}$.