Torus

Go to Surface Area or Volume.

Torus Facts

Notice these interesting things:

Torus Radii

  • 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

Torus Radii

Surface Area  = (2πR) × (2πr)
  = 4 × π2 × R × r

Example: r = 3 and R = 7

Surface Area = 4 × π2 × R × r
 = 4 × π2 × 7 × 3
 = 4 × π2 × 21
 = 84 × π2
 ≈ 829

The formula is often written in this shorter way:

Surface Area = 4π2Rr

Volume

Volume 
= (2πR) × (πr2)
  = 2 × π2 × R × r2

Example: r = 3 and R = 7

Volume = 2 × π2 × R × r2
 = 2 × π2 × 7 × 32
 = 2 × π2 × 7 × 9
 = 126 π2
 ≈ 1244

The formula is often written in this shorter way:

Volume = 2π2 Rr2

 

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 2πR):

Torus to Cylinder

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.

Torus to Cylinder Area

 

Torus Cushion Illustration

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)

The Volume and Area calculations will not work with this cushion because there is no hole.

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 Tire to a Donut:

Torus TireTorus Donut