Solid generated by an equilateral triangle about one side

Text

An equilateral triangle of side $6$ cm makes a full turn about one of its sides. Find the volume and the surface area of the resulting solid.

Solution

The vertex opposite the axis of rotation traces a circle of radius equal to the height $\rho=\dfrac{6\sqrt3}{2}=3\sqrt3$; the foot of the height falls at the midpoint of the side. Two equal cones result, of radius $\rho=3\sqrt3$, height $3$ and slant height $6$ (the triangle’s sides). Volume (two cones): $V=2\cdot\frac13\pi\rho^2\cdot3=2\cdot\frac13\pi\cdot27\cdot3=54\pi\approx169.65\ \text{cm}^3.$ Surface (two lateral surfaces): $S=2\cdot\pi\rho\cdot6=2\pi\cdot3\sqrt3\cdot6=36\sqrt3\,\pi\approx195.87\ \text{cm}^2.$ $\boxed{V=54\pi\approx169.65\ \text{cm}^3;\ S=36\sqrt3\,\pi\approx195.87\ \text{cm}^2}$