Volume of a Torus
Volume of a Torus
Using the definition of volume, we can calculate the sum of unit squares for any object with a closed space. In the case of a torus, a torus has a defined radius.
In the simulation below, you can input an arbitrary radius of your choice. To calculate the volume you can count each individual unit cubes to get the volume of a torus. Although we would get the correct answer, this is not practical as a torus with a large enough radius would take a long time to count up all the individual unit cubes each of length, width and height 1m by 1m by 1m.
Since a torus is a three dimensional representation of a circle, and to plot the two radii, one of the major radius in the xy plane and one of the minor radius in the z plane require three dimensions, we can assume that a torus will be in three dimensions. Since every torus has a radius major R, and a minor radius r, we can take the integral of the radius minor, r, to find the volume of a torus.
We can first start out by writing the equation of a circle:
Here we are assuming that the torus is obtained by rotating the circular region x2 + (y - R)2 = r2.
\[ x^2 + (y - R)^2 = r^2 \]
Solving for y we get:
\[ (y - R)^2 = r^2 - x^2 \]
\[ \sqrt{(y - R)^2} = \sqrt{r^2 - x^2} \]
\[ y - R = \sqrt{r^2 - x^2} \]
\[ y = \pm \sqrt{r^2 - x^2} + R \]
We can now take the integral of the area of a circle, and replacing r with y, we can then calculate the integral and find the volume of a torus:
\[ A = \pi r^2 \]
\[ V = \int_{-r}^{r} \pi r^2 dx \]
\[ V = \int_{-r}^{r} \pi y^2 dx \]
\[ V = \int_{-r}^{r} \pi (\sqrt{r^2 - x^2} + R)^2 - ( -\sqrt{r^2 - x^2} + R)^2 dx \]
\[ V = \int_{-r}^{r} \pi ((r^2 - x^2) + \sqrt{r^2 - x^2}R + R\sqrt{r^2 - x^2} + R^2) - ((r^2 - x^2) - \sqrt{r^2 - x^2}R - R\sqrt{r^2 - x^2} + R^2) dx \]
\[ V = \int_{-r}^{r} \pi ((r^2 - x^2) - (r^2 - x^2) + \sqrt{r^2 - x^2}R + \sqrt{r^2 - x^2}R + R\sqrt{r^2 - x^2} + R\sqrt{r^2 - x^2} + R^2 - R^2) dx \]
\[ V = \int_{-r}^{r} \pi (0 + 4\sqrt{r^2 - x^2}R + 0) dx \]
\[ V = \int_{-r}^{r} \pi 4\sqrt{r^2 - x^2}R dx \]
\[ V = \pi 4 R \left(\frac{1}{2} \pi r^2\right) \]
\[ V = 2 \pi R \pi r^2 \]
\[ V = (2 \pi R) (\pi r^2) \]
An equation we can create therefore to calculate the volume of a torus is the major radius times π times 2 all multiplied by π times the minor radius squared.
\[ V = (2 \pi R) (\pi r^2) \]
Since a torus is an enclosed space in three dimensions, that would mean that the units would be meters cubed.
\[ m^3 \]
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