Hi all,(adsbygoogle = window.adsbygoogle || []).push({});

First post here so 1) hello everyone, and 2) I have a problem that I am having trouble solving and was hoping to get some help from the folks on here.

I am trying to work out the formula for calculating the volume of a lozenge-shaped pond. I think this is basically a problem requiring some geometry and integration but it’s been a long time since I’ve done this sort of maths and so am a little rusty.

Basically the lozenge-shaped pond consists of an upper and lower surface that each have perfect semi-circular ends of radii r1 and r2 and each also have a middle section that is either square or rectangular. I’ve uploaded sketches of the pond to my website (see links) which should help clarify what I mean.

http://www.sudsolutions.com/misc/pond1.JPG [Broken]

http://www.sudsolutions.com/misc/pond2.JPG [Broken]

Both the upper and lower sections of the lozenge have the same shape but the lower section is smaller and would have sloping sides going down from the top. The information available would be the internal and external length of the pond as well as the internal and external width (which is also equal to the radii r1 and r2 respectively). The depth of the pond would also be known and has being denoted ‘d’ in the diagrams.

So I need to come up with a formula for working out the volume of any lozenge-shaped pond based on the information above. I think this involves:

1) Coming up with two formulas: one for the area of the lower surface and one for the area of the upper

2) Taking these as limits and then integrating to come up with an equation that gives the volume

For the lower surface I have:

http://www.sudsolutions.com/misc/upper_area.JPG [Broken]

And for the upper surface I have:

http://www.sudsolutions.com/misc/lower_area.JPG [Broken]

Now in order to come up with a single equation that gives the total volume of the shape I am guessing that I need to integrate these two equations, taking the lower surface area as the lower limit and the upper surface area as the upper limit, but also need to somehow work the variable “depth, d” into the equation as well.

Problem is I am stumped as how to do this, so any help here would be greatly appreciated!

Many thanks

-Rob

*edit* apologies for putting all the images as links but the tags don't seem to work in this section of the forum.

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# Volume of a lozenge shaped pond

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