# 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.

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
$$\quad r_2 = R + d/2$$    (I)
$$\quad r_1 = R - d/2$$    (II)
Add the above equations and simplify to obtain
$$\quad r_2 + r_1 = 2 R$$
which gives
$$\quad R = \dfrac{r_2 + r_1}{2}$$
Subtract the two equations (I) and (II) above and simplify top obtain
$$\quad r_2 - r_1 = d/2 + d/2 = d$$

The volume $$V$$ of the torus may be calculated as the volume of the cylinder in figure 3. Hence
$$\quad V = \pi \left( \dfrac{d}{2} \right)^2 \times 2 \pi R$$

Substitute $$d$$ and $$R$$ by their expressions in terms of $$r_1$$ and $$r_2$$ to obtain
$$\quad V = \pi \left( \dfrac{r_2 - r_1}{2} \right)^2 \times 2 \pi \left(\dfrac{r_2 + r_1}{2} \right)$$
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
$$\quad A_L = \pi d \times 2 \pi R$$
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: $$r_1$$ = 3 Outer Radius: $$r_2$$ = 5 Number of Decimals = 3