# Volume and Surface Area of Torus Calculator

## Formulas for the Volume and Surface Area

A torus ring is generated by rotating a small circle of diameter \( d \) along the perimeter of a larger circle. Views of a torus cut along the plane \( xy \; (z = 0) \) (on right) and another cut along the plane \( yz \; (x = 0) \) (on left) are shown below.\( r_1 \) is the inner radius and \( r_2 \) is the outer radius of the torus. A torus cut and developed into a cylinder of diameter \( d \) and length \( 2 \pi R \)

Let

\( r_2 = R + d/2 \) (I)

\( r_1 = R - d/2 \) (II)

Add the above equations and simplify to obtain

\( r_2 + r_1 = 2 R \)

which gives

\( R = \dfrac{r_2 + r_1}{2} \)

Subtract the two equations (I) and (II) above and simplify top obtain

\( r_2 - r_1 = d/2 + d/2 = d \)

The volume \( V \) of the torus may be calculated as the volume of the cylinder in figure 3. Hence

\( V = \pi \left( \dfrac{d}{2} \right)^2 \times 2 \pi R \)

Substitute \( d \) and \( R \) by their expressions in terms of \( r_1 \) and \( r_2 \) to obtain

\( V = \pi \left( \dfrac{r_2 - r_1}{2} \right)^2 \times 2 \pi \left(\dfrac{r_2 + r_1}{2} \right) \)
Simplify to obtain the formula
\[ \Large \color{red}{V = \dfrac{1}{4} \pi^2 ( r_2 - r_1 )^2 (r_2 + r_1)} \]

Using figure 3 above, the lateral surface area \( A_L \) of the torus may be calculated as the lateral surface area of the cylinder as follows

\( A_L = \pi d \times 2 \pi R \)

Substitute \( d \) and \( R \) by their expressions in terms of \( r_1 \) and \( r_2 \) to obtain

\[ \Large \color{red}{A_L = \pi (r_2 - r_1)(r_2 + r_1)} \]

## How to use the calculator

Enter the inner and outer radii of the torus, \( r_1 \) and \( r_2 \) respectively, as positive real numbers, with \( r_2 > r_1 \) and press "calculate". The outputs are volume \( V \) and the lateral area \(A_L \) of the torus.## Outputs

## More References and links

Sectors and Circles Problems .Circles, Sectors and Trigonometry Problems with Solutions and Answers .

Online Geometry Calculators and Solvers .