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 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)

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