Calculate volume for a plant pot

[Corey]

Hello I am trying to calculate the volume of a sloped or truncated container that has a different radius on the bottom and the top ie( a pot for plants ). There are many online calculators that I think use the following formula to solve. See pic.

However I would like to be able to calculate the volume when the pot is filled to any height (not necessarily to the top) . It would still be the same container so the top and bottom radius would not change but the third radius I wouldn’t be able to measure , but I would be able to measure the height , such as half or a third of the container height for example.

 

[Dr.Peterson]

Hello I am trying to calculate the volume of a sloped or truncated container that has a different radius on the bottom and the top ie( a pot for plants ). There are many online calculators that I think use the following formula to solve. See pic.

However I would like to be able to calculate the volume when the pot is filled to any height (not necessarily to the top) . It would still be the same container so the top and bottom radius would not change but the third radius I wouldn’t be able to measure , but I would be able to measure the height , such as half or a third of the container height for example.

The radius at the intermediate height will be a linear function of the height. That is, it increases at a constant rate, which you can find from the given radii and height.

 

[Corey]

Thanks Dr. Peterson. I’m not sure how to solve for this though. Would I just enter one of the known radius of the container into the attached formula with whatever height I want to get and the volume of the full container to get the new radius ? Then use the same formula to solve for volume with that new radius and height ?

 

[Dr.Peterson]

Thanks Dr. Peterson. I’m not sure how to solve for this though. Would I just enter one of the known radius of the container into the attached formula with whatever height I want to get and the volume of the full container to get the new radius ? Then use the same formula to solve for volume with that new radius and height ?

Do you understand what I said about a linear function? I can't tell what level of help you need, so I was hoping you'd try something and I could see how much you understand. (I don't want to do more for you than you need, so you gain as much experience as you can.)

Suppose you have an intermediate height, to which the pot is filled, and call it h_i. Then you can express the radius at that height, r_i, in terms of the known r₁, r₂, h, and h_i, which is the equation of a straight line with slope (r₂-r₁)/h. Plug that expression into the volume formula (using r_i and h_i in place of r₂ and h), and you'll have the formula you want.

 

[Corey]

I’m not sure I got it right. This volume doesn’t seem right.

 

[Dr.Peterson]

I’m not sure I got it right. This volume doesn’t seem right.

Can you explain why, here,

you replaced h with (r2-r1)/h2, rather than just with h2? And how is r2^2 equal to (r2-r1)^2/h2?

Let's back up, and focus on the formula for r3, without touching volume yet. For your specific numbers, what do you get for r3, the radius at height h2? Does the number you get make sense?

Then, to correct that formula, think about the equation of a line that passes through the points (0, r1) and (h1, r2). That's the linear function we're looking for.