Our Torus Surface Area Calculator will help you figure out how big a torus is for a different kind of set of radii. A torus is also a life-saving tube or ring, commonly known as a rescue buoy. In mathematics and science, the torus is also significant. Continue reading to learn what a torus is and how to calculate its surface area.

Meanwhile, see other related posts such as Torus Volume Calculator, Surface Area of a Cone, or maybe Surface Area of a Rectangular Prism Calculator. Also, check out this Byte Conversion as well.

Torus – Definition

We can define torus, from a mathematics point of view, as a doughnut-shaped object, such as an O ring. Also, it’s an object’s surface created by rotating a circle in three-dimensional space around an axis in the same plane as the circle. The surface produces a ring shape known as ring torus or simply torus if the axis of rotation does not contact the circle.

Additionally, the ring becomes a horn as the distance from the axis of rotation decreases. Swim rings or inner tubes are real-world items that resemble a torus form. Toric lenses are eyeglass lenses that incorporate cylindrical and spherical adjustments.


A torus is not to be mistaken for a solid torus, which is created by rotating a disk around an axis rather than a circle. A solid torus consists of a torus plus the volume contained inside it. O-rings, non-inflatable lifebuoys, ring doughnuts, and bagels are examples of real-world items that resemble a solid torus.

Attach the top edge to the bottom edge and the left edge to the right edge of a rectangular strip of flexible material, such as a rubber sheet, without any half-twists to make a torus.

Different Types of Torus

Depending on the relative sizes of a and c, three types of torus known as standard torus are possible. If there are no specifications, assume the torus form to be ring torus.

  • Create the ring torus when c > a.
  • When c = a, which is tangent to itself at the point, this is the creation of horn torus (0,0,0).
  • The self – intersecting spindle torus is formed when c < a

Surface Area of a Torus Formula

In addition to these radii, the torus can be represented as two radii, such as the inner (a) and outer (b) radii of the torus. That is, mathematically:

a = R - r
b = R + r

The torus’s surface area A is as follows:

A = 4 \timesπ^2\times r \times R

We can also use inner and outer radii to express the surface area:

A = π^2 \times (b - a) \times (b + a)

For example, the calculator uses the above equation to get the surface area where:

r = \frac {b - a} { 2}
R = \frac {a+b} { 2}

Torus Surface Area Calculator – How to Use?

So when we want to calculate the surface area of a torus, we need to follow these three simple steps:

1. Input the torus’s inner radius, a.

2: Input the torus’s outer radius, b.

3: Also, the calculator will now return the surface area of a torus using the formula above.

Torus Surface Area Calculator – Example

The main given volume radius of a ring-shaped item is 20 millimeters, while the minor given volume radius is 10 millimeters. For that item, calculate the torus’ surface area and volume.


Major Radius = 20, 

Minor Radius = 10


Surface area and Volume of Torus?

Step 1: So firstly, we’ll figure out how big the object’s surface is. In the formula below, substitute the values:

Surface   Area = 4\timesπ^2\times R\times r \newline Surface \; Area = 4 \times 3.14159262 \times 20 \times 10  \newline Surface \; Area =7895.6832

Step 2: Secondly thing that we need to do is to calculate the volume of the torus. 

Volume = 2\timesπ^2\times R \times r^2 \newline Volume = 2 \times 3.14159262 \times 20 \times 102 \newline Volume = 39478.4163

What is a torus used for?

A tore or anchor ring in geometry. In architecture, a big convex molding with a semicircular profile or a profile with a similar curve is commonly employed as the lowest element of the base, above the plinth when one is present. The main difference between it and the astragal is its size; the astragal is significantly smaller. Tore is another name for it.

The more or less enlarged endpoint of a stem or floral axis upon which the floral organs are positioned; the receptacle of a flower; the more or less expanded extremity of a stem or floral axis upon which the floral organs are situated.

A smooth rounded ridge or elongated protuberance, as of a muscle, especially a tuber cinereum of the brain, or that section of the third ventricle’s floor that is extended downward to provide a constricted channel from the third ventricle’s chamber into the pituitary body’s hollow. A torus is a zoological term for a portion or organ that resembles a torus, such as the ventral parapodium of several annelids.


What is a 2-torus?

A torus is a revolving surface created by rotating a circle in three-dimensional space around an axis that is coplanar with the circle.

How many edges does a torus have?

It just has one surface. There are no edges or vertices on it.

What shape is a torus?

A torus, or donut form, is the shape of this ring. Long before our structures, nature created the form. For example, the magnetic field around human bodies, and the magnetic field around Earth, are shaped like a torus. According to some scientists, the cosmos is a rotating torus.

How do you calculate the surface area of a torus?

The surface area of a torus is computed by multiplying the cross-section circumference by the ring circumference.