How to calculate for the volume taken up by a partially filled sphere?

1. Apr 30, 2012

mesa

So here is something a little different than the last couple formulas I needed, this one has to calculate for the volume taken up by a partially filled sphere.

The actual volume I need is at a depth of 1/2r but it would be better (and more fun) to build a formula that can be used for any given depth up to value r.

Any Thoughts?

2. Apr 30, 2012

Staff: Mentor

Have you had integral calculus? That is a pretty standard problem to solve with an integral. You add up the small volumes of lots of horizontal disks centered on the vertical axis. You know the horizontal radius of each disk, and the vertical height of each disk is dz tall. You integrate up the volumes of all of the very thin (dz vertical thickness) disks using an integral.

3. May 1, 2012

mesa

That's cheating ;)

4. May 2, 2012

mesa

Well, here is what I came up with so far. I decided to start with the specific volume (r/2) that I needed and go from there.

I drew out a "sphere" and decided to calculate the radius for the circle that represents the top of the volume using some trig. r×cos60=r/2 so that gave me the angle to radius~b on the sphere and gave a right triangle with the short side being the radius~b I was looking for, so used sin60 to get:

r~b = sin60 × r~a or
r~b = .866 × r~a

From here I decided to cut this thing up into cylinders of thickness determined by incremental angles of cos as a function of the radius and then add the stack up:

∏×(r~a×sin 60)^2×[r~a×cos60-r~a×cos59)] + ∏(r~a×sin59)^2×[r~a×cos59-r~a×cos58)] + ...

Now I need to find a function for the summation to sin0 although, to increase volume accuracy, it would be better to have the formula incorporate much thinner cylinders using a shorter function of height based off of cos. I know I don't need trig to represent the height but I think it will make it easier to come up with a formula using similar terms.

Any thoughts?

5. May 9, 2012

mesa

Okay, so I was told that the way I went about doing this is similar to calc so I tossed it out and tried something else. I think I have it, sort of...

So I started out with trying to find the area of a partially filled circle and then try to add the 3rd dimension to that formula.

I ended up getting (∏r^2)/2 - r^2[2∏(θ/360) + (sinθ)(cosθ)]
θ is a function of r based on the height of the volume the angle of the radius from the center to the top of the volume off of the ceneter horizontal diameter.

I then divided the formula for the volume of a sphere by the area of a circle and got 4/3r and multiplied the above forumal by that and it seems to work, Neat!

Now here comes the sort of lol. I need to convert θ into a function of radius (not a big deal) and this forumla can only calculate for volumes under 1/2 the total filled volume (this might be tough).

Any thoughts?

6. May 9, 2012

morrobay

What would that integral be ?

7. May 9, 2012

mathwonk

the derivative of the volume function is the area of the top of the portion of volume. hence the volume function is the antiderivative of that area function. you only need pythagoras to compute that area.

8. May 10, 2012

mesa

Can you show me how that works?

9. May 13, 2012

mesa

Need to figure out the Area taken up by a partially filled sphere

Okay, so now I need an equation that can figure out the surface area of a sphere that is taken up by whatever volume is filled. I believe I have the correct formula now but need a check on it. I got an area of 18.3 m^2 for a sphere with a diameter of 20m filled to 5m.

So how far off am I lol? :)

10. May 13, 2012

Vorde

Re: Need to figure out the Area taken up by a partially filled sphere

Well I got something completely different when I did it, but I can't help without knowing your process.

Would you mind walking us through your solution?

11. May 13, 2012

HallsofIvy

Staff Emeritus
Re: Need to figure out the Area taken up by a partially filled sphere

What do you mean by this? Are you talking about the surface area of the part of the sphere from the bottom up to 5 m above the bottom? I get far more than that. The entire surface area of a sphere of radius 20 m is $4\pi (400)= 5026$. Your answer would be only 18.3/5026= 0.3% of the whole sphere. That's much too small.

12. May 13, 2012

coolul007

Re: Need to figure out the Area taken up by a partially filled sphere

According to CRC, V=(1/3)(Pi)(h^2)(3R - h) = (1/6)(Pi)h(3a^2 + h^2) where R = Radius of sphere, a = radius of circle of partial fill, h = (depth of substance in sphere)perpendicular distance from center of circle to sphere surface.

S = 2(Pi)Rh = (Pi)Dh = (Pi)p^2, where D = diameter of sphere, and p = square root of(a^2 + h^2)

13. May 13, 2012

mesa

Re: Need to figure out the Area taken up by a partially filled sphere

Hah hah, yup I goofed this one pretty good, the formula should at least kind of work I thought, let me recrunch and see what the heck I did wrong :)

14. May 13, 2012

mesa

Re: Need to figure out the Area taken up by a partially filled sphere

I basically took an equation I had for figuring the volume of a partially fill sphere at r=11 and then subtracted from the same formula r=10 leaving a shell of a partial sphere volume 1 m thick so divided by 1m and got something probably wrong but hopefully ball park.

Okay I see what I did wrong, I forgot to add the third dimension to the equation, whoops! With that the answer from this formula spits out 386.5m^2 which is about 7.5% so it still seems low but a good deal better than .3% :)

Those CRC books have everything lol. Okay this should work as a check, thanks for the info!

15. May 13, 2012

Vorde

Re: Need to figure out the Area taken up by a partially filled sphere

Yeah your original attempt doesn't work (although I can see why it made sense at the time), using the other method will probably solve your question. I did it with calculus, but you shouldn't need it.

16. May 15, 2012

mesa

Re: Need to figure out the Area taken up by a partially filled sphere

Dang it, I keep getting ≈ 385 m^2, kind of close but not really. I think I'm gonna scrap what I've been trying and go with a different method.

What are your thoughts on this?

17. May 15, 2012

Vorde

Re: Need to figure out the Area taken up by a partially filled sphere

Are you comfortable with integral calculus? If not someone else will have to help you, I'm sure there is a general formula for a partially-filled sphere but I don't know it.

18. May 15, 2012

mesa

Re: Need to figure out the Area taken up by a partially filled sphere

I figured out the 'volume' formula for a partially filled sphere, but am now trying to get the 'area' taken up by that volume.

The 'area' thread was mistakenly combined with this 'volume' thread because one of the admins thought I was posting the same thread twice (the titles are worded nearly identically so I can see where the confusion came from).

I wrote to the PF admins to get it fixed and re-separate the two threads since they are completely different questions and an area question does not belong in a volume thread.

19. May 15, 2012

mesa

Re: Need to figure out the Area taken up by a partially filled sphere