Volume and Surface Area of Torus Calculator

An online calculator to calculate the volume and surface area of a torus ring is presented.

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.
torus ring
Figure 1. Torus Ring
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.
torus cut along the vertical axis
Figure 2. Cuts of Torus Ring
A torus cut and developed into a cylinder of diameter \( d \) and length \( 2 \pi R \)
torus cut cylinder
Figure 3. Torus Ring Developed as a Cylinder

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.

Inner Radius: \( r_1 \) =
Outer Radius: \( r_2 \) =
Number of Decimls =

Outputs


More References and links

Sectors and Circles Problems .
Circles, Sectors and Trigonometry Problems with Solutions and Answers .
Online Geometry Calculators and Solvers .

privacy policy