# 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 = 4 × **π**^{2} × R × r

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

**π**^{2}× R × r

**π**^{2}× 7 × 3

**π**^{2}× 21

**π**^{2}

The formula is often written in this shorter form:

Surface Area = 4**π**^{2} Rr

## Volume

Volume = 2 × *π*^{2} × R × r^{2}

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

**π**^{2}× R × r

^{2}

**π**^{2}× 7 × 3

^{2}

**π**^{2}× 7 × 9

**π**^{2}

The formula is often written in this shorter form:

Volume = 2*π*^{2} Rr^{2}

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

## Like a Cylinder

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.

And 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:**