1. **State the problem:** We compare two square pyramids. The first pyramid has height $h_1$ equal to one third its width $w_1$, i.e., $h_1 = \frac{1}{3} w_1$. The second pyramid has height $h_2$ equal to its width $w_2$, i.e., $h_2 = w_2$. We want to understand the relationship between these pyramids. 2. **Formula and rules:** For a square pyramid, the volume $V$ is given by $$V = \frac{1}{3} \times \text{base area} \times \text{height} = \frac{1}{3} w^2 h$$ where $w$ is the width of the square base and $h$ is the height. 3. **Express volumes:** - For the first pyramid: $$V_1 = \frac{1}{3} w_1^2 h_1 = \frac{1}{3} w_1^2 \times \frac{1}{3} w_1 = \frac{1}{9} w_1^3$$ - For the second pyramid: $$V_2 = \frac{1}{3} w_2^2 h_2 = \frac{1}{3} w_2^2 \times w_2 = \frac{1}{3} w_2^3$$ 4. **Compare volumes assuming equal widths:** If we assume $w_1 = w_2 = w$, then $$\frac{V_1}{V_2} = \frac{\frac{1}{9} w^3}{\frac{1}{3} w^3} = \frac{1}{9} \times \frac{3}{1} = \frac{1}{3}$$ So the first pyramid's volume is one third the volume of the second pyramid when widths are equal. 5. **Summary:** The pyramid with height one third its width has volume one third that of the pyramid whose height equals its width, assuming the base widths are the same. Final answer: The volume ratio is $\boxed{\frac{1}{3}}$ when widths are equal.