# Torus

*Go to Surface Area or Volume.*

Torus Facts

**Notice these interesting things:**

- It can be made by revolving a

small circle (radius**r**) along a line made

by a bigger circle (radius**R**). - It has no edges or vertices
- It is
**not**a polyhedron

*.*

Torus in the Sky

Torus in the Sky

The Torus is such a beautiful solid,

this one would be fun at the beach !

## Surface Area

Surface Area | = (2πR) × (2πr) |

= 4 × π^{2} × R × r |

### Example: r = 3 and R = 7

Surface Area = 4 × π

^{2}× R × r = 4 × π

^{2}× 7 × 3 = 4 × π

^{2}× 21 = 84 × π

^{2} ≈ 829

The formula is often written in this shorter way:

Surface Area = 4π^{2}Rr

## Volume

Volume |
= (2πR) × (πr^{2}) |

= 2 × π^{2} × R × r^{2} |

### Example: r = 3 and R = 7

Volume = 2 × π

^{2}× R × r^{2} = 2 × π

^{2}× 7 × 3^{2} = 2 × π

^{2}× 7 × 9 = 126 π

^{2} ≈ 1244

The formula is often written in this shorter way:

Volume = 2π^{2} Rr^{2}

*Note: Area and volume formulas only work when the torus has a hole!*

## Like a Cylinder

**Volume**: the volume is the same as if we "unfolded" a torus into a cylinder (of length 2*π*R):

As we unfold it, what gets lost from the outer part of the torus is perfectly balanced by what gets gained in the inner part.

**Surface Area**: the same is true for the surface area, not including the cylinder's bases.

And did you know that

*Torus*was the Latin word for a

**cushion**?

(This is not a *real* roman cushion, just an illustration I made)

When we have more than one torus they are called **tori**

## More Torus Images

As the small radius (**r**) gets larger and larger, the torus goes from looking like a

*to a*

**Tire**

**Donut:**