Mixing
Spherical Sector Volume
0
$begingroup$
I'm trying to find the volume of a spherical sector without knowing the height of the cap. Wikipedia provides this formula:
And says:
"where φ is half the cone angle, i.e., the angle between the rim of the cap and the direction to the middle of the cap as seen from the sphere center."
http://en.wikipedia.org/wiki/Spherical_sector
This sentence is kind of ambiguous and I was wondering if someone could just clarify it for me. Should Phi be inputted as the angle between the base of the cap and the side of the sector divided by 2?
geometry 3d
asked Oct 29 '14 at 2:59
VarrickVarrick
101
$endgroup$
add a comment |
0
$begingroup$
I'm trying to find the volume of a spherical sector without knowing the height of the cap. Wikipedia provides this formula:
And says:
"where φ is half the cone angle, i.e., the angle between the rim of the cap and the direction to the middle of the cap as seen from the sphere center."
http://en.wikipedia.org/wiki/Spherical_sector
This sentence is kind of ambiguous and I was wondering if someone could just clarify it for me. Should Phi be inputted as the angle between the base of the cap and the side of the sector divided by 2?
geometry 3d
asked Oct 29 '14 at 2:59
VarrickVarrick
101
$endgroup$
$begingroup$
On further inspection, the formula seems wrong regardless of what angle you put into Phi. Does anyone know a way of working out the volume of the spherical sector without knowing the height of the cap. Or indeed finding the height of the cap?
$endgroup$
– Varrick
Oct 29 '14 at 3:19
add a comment |
0
0
0
$begingroup$
I'm trying to find the volume of a spherical sector without knowing the height of the cap. Wikipedia provides this formula:
And says:
"where φ is half the cone angle, i.e., the angle between the rim of the cap and the direction to the middle of the cap as seen from the sphere center."
http://en.wikipedia.org/wiki/Spherical_sector
This sentence is kind of ambiguous and I was wondering if someone could just clarify it for me. Should Phi be inputted as the angle between the base of the cap and the side of the sector divided by 2?
geometry 3d
asked Oct 29 '14 at 2:59
VarrickVarrick
101
$endgroup$
I'm trying to find the volume of a spherical sector without knowing the height of the cap. Wikipedia provides this formula:
And says:
"where φ is half the cone angle, i.e., the angle between the rim of the cap and the direction to the middle of the cap as seen from the sphere center."
http://en.wikipedia.org/wiki/Spherical_sector
This sentence is kind of ambiguous and I was wondering if someone could just clarify it for me. Should Phi be inputted as the angle between the base of the cap and the side of the sector divided by 2?
geometry 3d
geometry 3d
asked Oct 29 '14 at 2:59
VarrickVarrick
101
asked Oct 29 '14 at 2:59
VarrickVarrick
101
asked Oct 29 '14 at 2:59
VarrickVarrick
101
asked Oct 29 '14 at 2:59
VarrickVarrick
101
asked Oct 29 '14 at 2:59
VarrickVarrick
101
101
$begingroup$
On further inspection, the formula seems wrong regardless of what angle you put into Phi. Does anyone know a way of working out the volume of the spherical sector without knowing the height of the cap. Or indeed finding the height of the cap?
$endgroup$
– Varrick
Oct 29 '14 at 3:19
add a comment |
$begingroup$
On further inspection, the formula seems wrong regardless of what angle you put into Phi. Does anyone know a way of working out the volume of the spherical sector without knowing the height of the cap. Or indeed finding the height of the cap?
$endgroup$
– Varrick
Oct 29 '14 at 3:19
$begingroup$
On further inspection, the formula seems wrong regardless of what angle you put into Phi. Does anyone know a way of working out the volume of the spherical sector without knowing the height of the cap. Or indeed finding the height of the cap?
$endgroup$
– Varrick
Oct 29 '14 at 3:19
$begingroup$
On further inspection, the formula seems wrong regardless of what angle you put into Phi. Does anyone know a way of working out the volume of the spherical sector without knowing the height of the cap. Or indeed finding the height of the cap?
$endgroup$
– Varrick
Oct 29 '14 at 3:19
add a comment |
1 Answer
1
active
oldest
votes
0
$begingroup$
$2phi$ is the angle of the cone (spherical sector)
From Wikipedia:
$V=frac{2pi r^2 h}{3}$ ...1
From the cone:
$frac{r-h}{r}=cosphi$
After simplifying
$h=r(1-cosphi)$
Replace in 1 to get
$V=frac{2pi r^3}{3}(1-cosphi)$
answered Oct 29 '14 at 3:28
Fahd SiddiquiFahd Siddiqui
1328
$endgroup$
$begingroup$
I'm pretty confused, I just tried both of these formulas to calculate height and got completely different answers... upload.wikimedia.org/math/a/9/a/… I put the radius as 1 and theta as 90 and got for the first one: 0.47467801118227027, and the second one: 0.14909647546588156
$endgroup$
– Varrick
Oct 29 '14 at 3:39
$begingroup$
Can you demonstrate some of your working?
$endgroup$
– Fahd Siddiqui
Oct 29 '14 at 3:41
$begingroup$
Sure, it's in computer code. Hopefully it's not too difficult to work out if you've never seen code before. It maybe easier to read if you copy and paste into notepad and space it onto seperate lines. l is radius of circle (or length of side of cone). h is height of cone. h = l * Math.sin(theta); r = sqrt(Math.pow(l,2) - (h^2)); cone volume = pi * (r^2) * (h^3); cap height (method one) = l * (1 - cos(theta/2)); cap height (method two) = l - sqrt((l^2) - ((r*2^2)/4)); scv = ((2 * pi * l^2)/3) * cap height; csv = scv - cv; // circular segment volume
$endgroup$
– Varrick
Oct 29 '14 at 4:13
$begingroup$
So basically, if I use the two formula's for height on this page: en.wikipedia.org/wiki/Circular_segment. And enter theta as 90, R as 1, c as (2*R*Cos(45) I get for the first equation... 0.47467801118227027 and for the second... 0.14909647546588156
$endgroup$
– Varrick
Oct 30 '14 at 4:23
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
draft saved
draft discarded
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f996093%2fspherical-sector-volume%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
0
$begingroup$
$2phi$ is the angle of the cone (spherical sector)
From Wikipedia:
$V=frac{2pi r^2 h}{3}$ ...1
From the cone:
$frac{r-h}{r}=cosphi$
After simplifying
$h=r(1-cosphi)$
Replace in 1 to get
$V=frac{2pi r^3}{3}(1-cosphi)$
answered Oct 29 '14 at 3:28
Fahd SiddiquiFahd Siddiqui
1328
$endgroup$
$begingroup$
I'm pretty confused, I just tried both of these formulas to calculate height and got completely different answers... upload.wikimedia.org/math/a/9/a/… I put the radius as 1 and theta as 90 and got for the first one: 0.47467801118227027, and the second one: 0.14909647546588156
$endgroup$
– Varrick
Oct 29 '14 at 3:39
$begingroup$
Can you demonstrate some of your working?
$endgroup$
– Fahd Siddiqui
Oct 29 '14 at 3:41
$begingroup$
Sure, it's in computer code. Hopefully it's not too difficult to work out if you've never seen code before. It maybe easier to read if you copy and paste into notepad and space it onto seperate lines. l is radius of circle (or length of side of cone). h is height of cone. h = l * Math.sin(theta); r = sqrt(Math.pow(l,2) - (h^2)); cone volume = pi * (r^2) * (h^3); cap height (method one) = l * (1 - cos(theta/2)); cap height (method two) = l - sqrt((l^2) - ((r*2^2)/4)); scv = ((2 * pi * l^2)/3) * cap height; csv = scv - cv; // circular segment volume
$endgroup$
– Varrick
Oct 29 '14 at 4:13
$begingroup$
So basically, if I use the two formula's for height on this page: en.wikipedia.org/wiki/Circular_segment. And enter theta as 90, R as 1, c as (2*R*Cos(45) I get for the first equation... 0.47467801118227027 and for the second... 0.14909647546588156
$endgroup$
– Varrick
Oct 30 '14 at 4:23
add a comment |
0
$begingroup$
$2phi$ is the angle of the cone (spherical sector)
From Wikipedia:
$V=frac{2pi r^2 h}{3}$ ...1
From the cone:
$frac{r-h}{r}=cosphi$
After simplifying
$h=r(1-cosphi)$
Replace in 1 to get
$V=frac{2pi r^3}{3}(1-cosphi)$
answered Oct 29 '14 at 3:28
Fahd SiddiquiFahd Siddiqui
1328
$endgroup$
$begingroup$
I'm pretty confused, I just tried both of these formulas to calculate height and got completely different answers... upload.wikimedia.org/math/a/9/a/… I put the radius as 1 and theta as 90 and got for the first one: 0.47467801118227027, and the second one: 0.14909647546588156
$endgroup$
– Varrick
Oct 29 '14 at 3:39
$begingroup$
Can you demonstrate some of your working?
$endgroup$
– Fahd Siddiqui
Oct 29 '14 at 3:41
$begingroup$
Sure, it's in computer code. Hopefully it's not too difficult to work out if you've never seen code before. It maybe easier to read if you copy and paste into notepad and space it onto seperate lines. l is radius of circle (or length of side of cone). h is height of cone. h = l * Math.sin(theta); r = sqrt(Math.pow(l,2) - (h^2)); cone volume = pi * (r^2) * (h^3); cap height (method one) = l * (1 - cos(theta/2)); cap height (method two) = l - sqrt((l^2) - ((r*2^2)/4)); scv = ((2 * pi * l^2)/3) * cap height; csv = scv - cv; // circular segment volume
$endgroup$
– Varrick
Oct 29 '14 at 4:13
$begingroup$
So basically, if I use the two formula's for height on this page: en.wikipedia.org/wiki/Circular_segment. And enter theta as 90, R as 1, c as (2*R*Cos(45) I get for the first equation... 0.47467801118227027 and for the second... 0.14909647546588156
$endgroup$
– Varrick
Oct 30 '14 at 4:23
add a comment |
0
0
0
$begingroup$
$2phi$ is the angle of the cone (spherical sector)
From Wikipedia:
$V=frac{2pi r^2 h}{3}$ ...1
From the cone:
$frac{r-h}{r}=cosphi$
After simplifying
$h=r(1-cosphi)$
Replace in 1 to get
$V=frac{2pi r^3}{3}(1-cosphi)$
answered Oct 29 '14 at 3:28
Fahd SiddiquiFahd Siddiqui
1328
$endgroup$
$2phi$ is the angle of the cone (spherical sector)
From Wikipedia:
$V=frac{2pi r^2 h}{3}$ ...1
From the cone:
$frac{r-h}{r}=cosphi$
After simplifying
$h=r(1-cosphi)$
Replace in 1 to get
$V=frac{2pi r^3}{3}(1-cosphi)$
answered Oct 29 '14 at 3:28
Fahd SiddiquiFahd Siddiqui
1328
answered Oct 29 '14 at 3:28
Fahd SiddiquiFahd Siddiqui
1328
answered Oct 29 '14 at 3:28
Fahd SiddiquiFahd Siddiqui
1328
answered Oct 29 '14 at 3:28
Fahd SiddiquiFahd Siddiqui
1328
1328
$begingroup$
I'm pretty confused, I just tried both of these formulas to calculate height and got completely different answers... upload.wikimedia.org/math/a/9/a/… I put the radius as 1 and theta as 90 and got for the first one: 0.47467801118227027, and the second one: 0.14909647546588156
$endgroup$
– Varrick
Oct 29 '14 at 3:39
$begingroup$
Can you demonstrate some of your working?
$endgroup$
– Fahd Siddiqui
Oct 29 '14 at 3:41
$begingroup$
Sure, it's in computer code. Hopefully it's not too difficult to work out if you've never seen code before. It maybe easier to read if you copy and paste into notepad and space it onto seperate lines. l is radius of circle (or length of side of cone). h is height of cone. h = l * Math.sin(theta); r = sqrt(Math.pow(l,2) - (h^2)); cone volume = pi * (r^2) * (h^3); cap height (method one) = l * (1 - cos(theta/2)); cap height (method two) = l - sqrt((l^2) - ((r*2^2)/4)); scv = ((2 * pi * l^2)/3) * cap height; csv = scv - cv; // circular segment volume
$endgroup$
– Varrick
Oct 29 '14 at 4:13
$begingroup$
So basically, if I use the two formula's for height on this page: en.wikipedia.org/wiki/Circular_segment. And enter theta as 90, R as 1, c as (2*R*Cos(45) I get for the first equation... 0.47467801118227027 and for the second... 0.14909647546588156
$endgroup$
– Varrick
Oct 30 '14 at 4:23
add a comment |
$begingroup$
I'm pretty confused, I just tried both of these formulas to calculate height and got completely different answers... upload.wikimedia.org/math/a/9/a/… I put the radius as 1 and theta as 90 and got for the first one: 0.47467801118227027, and the second one: 0.14909647546588156
$endgroup$
– Varrick
Oct 29 '14 at 3:39
$begingroup$
Can you demonstrate some of your working?
$endgroup$
– Fahd Siddiqui
Oct 29 '14 at 3:41
$begingroup$
Sure, it's in computer code. Hopefully it's not too difficult to work out if you've never seen code before. It maybe easier to read if you copy and paste into notepad and space it onto seperate lines. l is radius of circle (or length of side of cone). h is height of cone. h = l * Math.sin(theta); r = sqrt(Math.pow(l,2) - (h^2)); cone volume = pi * (r^2) * (h^3); cap height (method one) = l * (1 - cos(theta/2)); cap height (method two) = l - sqrt((l^2) - ((r*2^2)/4)); scv = ((2 * pi * l^2)/3) * cap height; csv = scv - cv; // circular segment volume
$endgroup$
– Varrick
Oct 29 '14 at 4:13
$begingroup$
So basically, if I use the two formula's for height on this page: en.wikipedia.org/wiki/Circular_segment. And enter theta as 90, R as 1, c as (2*R*Cos(45) I get for the first equation... 0.47467801118227027 and for the second... 0.14909647546588156
$endgroup$
– Varrick
Oct 30 '14 at 4:23
$begingroup$
I'm pretty confused, I just tried both of these formulas to calculate height and got completely different answers... upload.wikimedia.org/math/a/9/a/… I put the radius as 1 and theta as 90 and got for the first one: 0.47467801118227027, and the second one: 0.14909647546588156
$endgroup$
– Varrick
Oct 29 '14 at 3:39
$begingroup$
I'm pretty confused, I just tried both of these formulas to calculate height and got completely different answers... upload.wikimedia.org/math/a/9/a/… I put the radius as 1 and theta as 90 and got for the first one: 0.47467801118227027, and the second one: 0.14909647546588156
$endgroup$
– Varrick
Oct 29 '14 at 3:39
$begingroup$
Can you demonstrate some of your working?
$endgroup$
– Fahd Siddiqui
Oct 29 '14 at 3:41
$begingroup$
Can you demonstrate some of your working?
$endgroup$
– Fahd Siddiqui
Oct 29 '14 at 3:41
$begingroup$
Sure, it's in computer code. Hopefully it's not too difficult to work out if you've never seen code before. It maybe easier to read if you copy and paste into notepad and space it onto seperate lines. l is radius of circle (or length of side of cone). h is height of cone. h = l * Math.sin(theta); r = sqrt(Math.pow(l,2) - (h^2)); cone volume = pi * (r^2) * (h^3); cap height (method one) = l * (1 - cos(theta/2)); cap height (method two) = l - sqrt((l^2) - ((r*2^2)/4)); scv = ((2 * pi * l^2)/3) * cap height; csv = scv - cv; // circular segment volume
$endgroup$
– Varrick
Oct 29 '14 at 4:13
$begingroup$
Sure, it's in computer code. Hopefully it's not too difficult to work out if you've never seen code before. It maybe easier to read if you copy and paste into notepad and space it onto seperate lines. l is radius of circle (or length of side of cone). h is height of cone. h = l * Math.sin(theta); r = sqrt(Math.pow(l,2) - (h^2)); cone volume = pi * (r^2) * (h^3); cap height (method one) = l * (1 - cos(theta/2)); cap height (method two) = l - sqrt((l^2) - ((r*2^2)/4)); scv = ((2 * pi * l^2)/3) * cap height; csv = scv - cv; // circular segment volume
$endgroup$
– Varrick
Oct 29 '14 at 4:13
$begingroup$
So basically, if I use the two formula's for height on this page: en.wikipedia.org/wiki/Circular_segment. And enter theta as 90, R as 1, c as (2*R*Cos(45) I get for the first equation... 0.47467801118227027 and for the second... 0.14909647546588156
$endgroup$
– Varrick
Oct 30 '14 at 4:23
$begingroup$
So basically, if I use the two formula's for height on this page: en.wikipedia.org/wiki/Circular_segment. And enter theta as 90, R as 1, c as (2*R*Cos(45) I get for the first equation... 0.47467801118227027 and for the second... 0.14909647546588156
$endgroup$
– Varrick
Oct 30 '14 at 4:23
add a comment |
draft saved
draft discarded
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
draft saved
draft discarded
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f996093%2fspherical-sector-volume%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
$begingroup$
On further inspection, the formula seems wrong regardless of what angle you put into Phi. Does anyone know a way of working out the volume of the spherical sector without knowing the height of the cap. Or indeed finding the height of the cap?
$endgroup$
– Varrick
Oct 29 '14 at 3:19