description of the img
Inner radius (a)
Outer radius (b)
Tube radius (r)
Radius of revolution (R)
Surface area
Your torus is horn type.

Torus Surface Area Calculator

By Rahul Dhari
Last updated: Jun 14, 2021

This torus surface area calculator will assist you to estimate the surface area of a torus for a given pair of radii. You must have encountered this shape in daily life on your plate as a doughnut or a bagel, or on the roads underneath vehicles 😉. The life-saving tube or ring, aka the rescue buoy, is also a torus. The torus likewise has significance in mathematics and physics. Read on to understand what is a torus and how to calculate the surface area of torus?

What is a torus?

A torus is a 3D shape obtained by revolving a circle around an axis. This shape is commonly found in doughnuts, rings, tires, tubes. If you take a ring and circularly trace around a pencil, you get a torus. In modern design software, it is fairly easy to draw them by using a revolve command with a circle as a cross-section. A torus has two radii – the first radius is the radius of the cross-section r and the second radius R is the radius of revolution which is the distance between the center axis and the center of cross-section.

Any point on a torus is defined using a modified coordinate system having two directions – toroidal (red arrow) and poloidal (green arrow).

Directions on a torus - toroidal (along the bigger ring), and poloidal (along the smaller ring)
Torus directions

Based on the combinations of the two radii, we can obtain multiple types of tori. Such that:

  • Ring type (R > r)
  • Horn type (R = r)
  • Spindle type (R < r)

In addition to the these radii, the torus can also be expressed in the form of two radii such as inner (a) and outer radii (b) of the torus. Mathematically, that's:

a = R - r
b = R + r

The surface area A of the said torus is:

A = 4 * π2 * r * R
The surface area can also be written in terms of inner and outer radii:
A = π2 * ( b - a ) * (b + a)
The above equation is used in the calculator to determine the surface area where:
r = (b - a) / 2
R = (a + b) / 2

Note: This calculator is only applicable to ring type or horn type tori. Furthermore, in the case of horn type torus, i.e., R = r, minor radius, a becomes zero.

How to use the torus surface area calculator?

Follow three simple steps to find out the surface area of a torus.

  • Step 1: Enter the inner radius of torus, a.
  • Step 2: Enter the outer radius of torus, b.
  • Step 3: The calculator will now use the above formula to return the surface area of a torus.

Example: How to calculate surface area of a torus?

Find the surface area of a horn type torus having a radius of cross-section r = 1 m. Note: The torus is of horn type, i.e., r = R.

Let us first convert the radii into the inner and outer radius, a and b.

a = R - r = 0 m
b = R + r = 1 m
  • Step 1: Enter the inner radius of torus, a = 0 m.
  • Step 2: Enter the outer radius of torus, b = 1 m.
  • Step 3: The torus surface area calculator will now return:
V = π2 * ( b - a ) * (b + a)
V = π2 * (1 - 0) * (1 + 0) = 9.87 m2
The surface area for the torus is 9.87 m2. You might also be interested in determining the volume of a torus or its surface area to volume ratio.


What is a torus?

A torus is a 3D circular shape with a circle as a cross-section. The shape is commonly found in doughnuts, tires, and hoops. The shape is obtained when you revolve a circle along a circular path along an axis normal to the circle.

How is a torus formed?

A solid torus is formed when you trace a circle along another circle in the plane without any self intersection.

What is the equation of a torus?

The equation of a torus is (R - √(x2 + y2))2 + z2 = rc2. A point (x, y, z) on the torus can be represented using the equation.

How do you calculate surface area of a torus?

The surface area of a torus is calculated by multiplying the circumference of the cross-section by the circumference of the ring. Volume = 2 × π × rc × 2 × π × R.

Rahul Dhari
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