A torus is a spatial figure that can be described as a circle rotated around an axis in the same plane without touching the axis. The figure resembles a swim ring or a donut.

A torus is defined by two radii:

• Major radius \( \large R \): The distance from the center of the torus to the center of the tube.
• Minor radius \( \large r \): The radius of the tube itself.

Volume

The volume of a torus can be found by considering it as a circle with area \( \large \pi \cdot r^2 \), moving around a circle with circumference \( \large 2 \cdot \pi \cdot R \):

$$ \large V = ( \pi \cdot r^2 ) \cdot ( 2 \cdot \pi \cdot R ) $$

Thus we get:

$$ \large V = 2 \cdot \pi^2 \cdot R \cdot r^2 $$

Surface area

The surface area can be found by considering the torus as a circle with circumference \( \large 2 \cdot \pi \cdot r \), moving around a circle with circumference \( \large 2 \cdot \pi \cdot R \):

$$ \large S = ( 2 \cdot \pi \cdot r ) \cdot ( 2 \cdot \pi \cdot R ) $$

Thus we get:

$$ \large S = 4 \cdot \pi^2 \cdot R \cdot r $$