# Torus Surface Area Calculator

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 a 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 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 only applies to ring type or horn type tori. Furthermore, in the case of a 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.

## 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 = r꜀². 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 × r꜀ × 2 × π × R.