# 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). 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.

## FAQ

### 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 in
in
in
in
Surface area
in²
People also viewed…

### Circumference

Use this free circumference calculator to find the area, circumference and diameter of a circle.

### Fraction to decimal converter

Use the fraction to decimal converter to convert, well, fractions to decimals - give it 1/4, get 0.25.

### Free fall

Our free fall calculator can find the velocity of a falling object and the height it drops from.

### Linear combination

Linear combination calculator helps you solve a system of equations with the method of combining linear equations. 