Step 1: Identifying the Solid Formed
When the right triangle is revolved about the side \( BC = 12 \, \text{cm} \), a cone is formed. The radius of the cone will be \( AB = 5 \, \text{cm} \), and the height of the cone will be \( BC = 12 \, \text{cm} \).
Step 2: Volume of the Cone
The formula for the volume \( V \) of a cone is: \[ V = \frac{1}{3} \pi r^2 h \] where \( r \) is the radius and \( h \) is the height of the cone. Substituting the values \( r = 5 \, \text{cm} \) and \( h = 12 \, \text{cm} \): \[ V = \frac{1}{3} \pi (5)^2 \times 12 = \frac{1}{3} \pi \times 25 \times 12 = \frac{1}{3} \pi \times 300 = 100 \pi \, \text{cm}^3 \] Therefore, the volume of the solid obtained is \( 100 \pi \, \text{cm}^3 \).
The volume of the solid obtained by revolving the right triangle about the side \( BC = 12 \, \text{cm} \) is \( 100 \pi \, \text{cm}^3 \), which is approximately \( 314.16 \, \text{cm}^3 \) when using \( \pi \approx 3.1416 \).