Torus

A torus is a unique three-dimensional shape that looks like a donut. Swimming tubes and car or bike tubes are typical examples of the shape of a torus. The diagram below shows the shape of a torus.

A torus is not a polyhedron as it has no flat faces. It is curved throughout, and it has a hole.

The shape of a torus is formed by revolving a circle in the 3-D space about an axis coplanar to the circle. Another smaller circle is rotating around the bigger circle. The diagram below shows the formation of the shape.

The smaller circle is said to be the tube. The radius of the bigger circle is often said to be the major radius. In comparison, the radius of the smaller circle is the tube radius or minor radius

Formulas

Surface Area 

We calculate the surface area by multiplying the circumference of the bigger circle (with radius R) by, the smaller circle (with radius r):

⇒ Surface Area (SA) = 2πR × 2πr

The final formula is :

Find the surface area of a torus with an outer radius of 9 cm and a tube radius of 3 cm.

Solution:

As we know,
Surface Area (SA) = 4π²Rr, here R = 9 cm, r = 3 cm, π = 3.14
∴ SA = 4 × (3.14)² × 9 × 3
= 1064.84 cm²

Find the surface area of a torus with an outer radius of 7 cm and a tube radius of 11 cm.

Solution:

As we know,
Surface Area (SA) = 4π²Rr, here R = 7 cm, r = 11 cm, π = 3.14
∴ SA = 4 × (3.14)² × 7 × 11
= 3036.75 cm²

Volume

The formula is:

Volume (V) = 2πR × πr² = V = 2π²Rr²

If a torus is cut and unfolded, it will take the shape of a cylinder, as shown in the diagram below.

The volume of a cylinder is πr² × length; here the length equals 2πR, where R = major radius of the torus.

Therefore, volume of a cylinder = πr² × length, here length = 2πR

So the volume of a torus is actually 2π²Rr²

Find the volume of a torus with an outer radius of 12 cm and a tube radius of 5 cm.

Solution:

As we know,
Volume (V) = 2π²Rr², here R = 12 cm, r = 5 cm, π = 3.14
∴ V = 2 × (3.14)² × 12 × 5²
= 5915.76 cm³