A torus ring is generated by rotating a small circle of diameter \( d \) along the perimeter of a larger circle.
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
\[ \Large \color{red}{A_L = \pi (r_2 - r_1)(r_2 + r_1)} \]
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.
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