1. **State the problem:** We have two similar cylinders. The radius of the smaller cylinder is $r_1 = 5$ cm, and the radius of the larger cylinder is $r_2 = 6$ cm with volume $V_2 = 432$ cm³. We need to find the volume $V_1$ of the smaller cylinder. 2. **Formula and rules:** For similar solids, volumes scale as the cube of the ratio of corresponding linear dimensions. That is, $$\frac{V_1}{V_2} = \left(\frac{r_1}{r_2}\right)^3$$ 3. **Substitute known values:** $$\frac{V_1}{432} = \left(\frac{5}{6}\right)^3$$ 4. **Calculate the cube:** $$\left(\frac{5}{6}\right)^3 = \frac{5^3}{6^3} = \frac{125}{216}$$ 5. **Solve for $V_1$:** $$V_1 = 432 \times \frac{125}{216}$$ 6. **Simplify the multiplication:** $$432 = \cancel{216} \times 2$$ So, $$V_1 = \cancel{216} \times 2 \times \frac{125}{\cancel{216}} = 2 \times 125 = 250$$ 7. **Final answer:** The volume of the smaller cylinder is $$\boxed{250 \text{ cm}^3}$$