How to find volume and surface area of a spindle torus?
$begingroup$
I know that you can use the formulas described in Pappus' centroid theorem, detailed here. But does Pappus' centroid theorem hold true for all forms of a torus: ring, horn, and spindle? I found another website that uses Pappus' centroid theorem for the volume and surface area of a ring or horn torus, but a different formula for the spindle torus.
This is the formula listed for the spindle torus:
$$
V = frac23 pi ( 2r^2 + R^2 ) sqrt{r^2 - R^2} + pi r^2 R left[pi + 2arctanleft( frac{R}{sqrt{r^2 - R^2}} right) right]
$$
where $r$ is the minor radius and $R$ is the major radius.
Is this formula for the spindle torus' volume accurate, or do I just have to use Pappus' centroid theorem?
calculus area volume solid-of-revolution
edited Dec 10 '18 at 11:58
Brahadeesh
6,24242361
asked Dec 10 '18 at 11:46
fi12fi12
1113
$endgroup$
add a comment |
$begingroup$
I know that you can use the formulas described in Pappus' centroid theorem, detailed here. But does Pappus' centroid theorem hold true for all forms of a torus: ring, horn, and spindle? I found another website that uses Pappus' centroid theorem for the volume and surface area of a ring or horn torus, but a different formula for the spindle torus.
This is the formula listed for the spindle torus:
$$
V = frac23 pi ( 2r^2 + R^2 ) sqrt{r^2 - R^2} + pi r^2 R left[pi + 2arctanleft( frac{R}{sqrt{r^2 - R^2}} right) right]
$$
where $r$ is the minor radius and $R$ is the major radius.
Is this formula for the spindle torus' volume accurate, or do I just have to use Pappus' centroid theorem?
calculus area volume solid-of-revolution
edited Dec 10 '18 at 11:58
Brahadeesh
6,24242361
asked Dec 10 '18 at 11:46
fi12fi12
1113
$endgroup$
add a comment |
$begingroup$
I know that you can use the formulas described in Pappus' centroid theorem, detailed here. But does Pappus' centroid theorem hold true for all forms of a torus: ring, horn, and spindle? I found another website that uses Pappus' centroid theorem for the volume and surface area of a ring or horn torus, but a different formula for the spindle torus.
This is the formula listed for the spindle torus:
$$
V = frac23 pi ( 2r^2 + R^2 ) sqrt{r^2 - R^2} + pi r^2 R left[pi + 2arctanleft( frac{R}{sqrt{r^2 - R^2}} right) right]
$$
where $r$ is the minor radius and $R$ is the major radius.
Is this formula for the spindle torus' volume accurate, or do I just have to use Pappus' centroid theorem?
calculus area volume solid-of-revolution
edited Dec 10 '18 at 11:58
Brahadeesh
6,24242361
asked Dec 10 '18 at 11:46
fi12fi12
1113
$endgroup$
I know that you can use the formulas described in Pappus' centroid theorem, detailed here. But does Pappus' centroid theorem hold true for all forms of a torus: ring, horn, and spindle? I found another website that uses Pappus' centroid theorem for the volume and surface area of a ring or horn torus, but a different formula for the spindle torus.
This is the formula listed for the spindle torus:
$$
V = frac23 pi ( 2r^2 + R^2 ) sqrt{r^2 - R^2} + pi r^2 R left[pi + 2arctanleft( frac{R}{sqrt{r^2 - R^2}} right) right]
$$
where $r$ is the minor radius and $R$ is the major radius.
Is this formula for the spindle torus' volume accurate, or do I just have to use Pappus' centroid theorem?
calculus area volume solid-of-revolution
calculus area volume solid-of-revolution
edited Dec 10 '18 at 11:58
Brahadeesh
6,24242361
asked Dec 10 '18 at 11:46
fi12fi12
1113
edited Dec 10 '18 at 11:58
Brahadeesh
6,24242361
asked Dec 10 '18 at 11:46
fi12fi12
1113
edited Dec 10 '18 at 11:58
Brahadeesh
6,24242361
edited Dec 10 '18 at 11:58
Brahadeesh
6,24242361
edited Dec 10 '18 at 11:58
Brahadeesh
6,24242361
6,24242361
asked Dec 10 '18 at 11:46
fi12fi12
1113
asked Dec 10 '18 at 11:46
fi12fi12
1113
asked Dec 10 '18 at 11:46
fi12fi12
1113
1113
add a comment |
add a comment |
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