(a) Arc length \(=\frac{240}{360}(2\pi)(12)=16\pi\text{ cm}\).

(b) The arc becomes the cone base circumference: \(2\pi r=16\pi\), so r=8 cm. The sector radius becomes the slant height, 12 cm. Vertical height \(h=\sqrt{12^2-8^2}=\sqrt{80}=4\sqrt5\text{ cm}\). Hence \(V=\frac13\pi(8)^2(4\sqrt5)=\boxed{256\sqrt5\pi/3\text{ cm}^3}\).