How do you calculate the volume of an irregular object?

Question

i made this model of a hot air balloon. i know the perimeter of each shape i used because i carefully drafted the shapes i used. but i want to know how i can calculate the volume, so i can find out how much air it can hold. i want to know if there is a certain formula that you use. thanks and honest answers plz!!!!!!!!!!!!!!!!!!!!!!!

Answer 1

It would be possible to calculate it, but it requires integral calculus.

Here's the basic procedure: imagine you sliced the balloon horizontally into a lot of pieces. Those pieces would have a circumference equal to the width of the pieces at that point, multiplied by the number of pieces that make up the balloon. If you calculate the upper and lower circumferences, you can calculate conic sections that approximate the curved shape of the balloon. By adding up all of those conical approximations, you can get an approximation of the volume of the balloon. The more cones you calculate, the more accurate that approximation will be.

Answer 2

In practice I wouldn't think calculus is the best approach. A variation of Archimedes' method might be easier. Note that you shouldn't use the classic Archimedes method (get a container of water which has an overflow spout and is filled right to the brink of overflowing; submerge the balloon, collect the water that overflows and measure its volume). The reason is that the balloon is not rigid, and submerging it in water will make the volume decrease since the water pressure will be greater than the air pressure. So you will get inaccurate results. However, you could do something like covering the balloon in a very thin layer of plaster or papier-mache. Ideally, you should build a split line into this so you can retrieve the balloon after it has dried. Break the mould and take the balloon out, then re-seal the mould (except for the end where the balloon is tied) and fill it with water. Then tip the water into a measuring jug and you're done. But if you really want to do it using calculus, the easiest case is if the balloon has rotational symmetry (like a standard ballon shape). You'll need a function describing the radius of the balloon at any point along the rotation axis. If you don't have one, you can measure it at various points and derive an approximate result. If you have a function f(x), where x goes from 0 to h and f(x) is the radius at point x, the volume of the balloon is π∫(0 to h) f(x)^2 dx. How you calculate this depends very much on the form of f(x), so I won't go into it here. For measured values, the idea is that you divide the rotational axis up into small segments, measure the radius r in each segment, and calculate the approximate volume contributed by that segment as πr^2 multiplied by the length of that segment. Then you add all these values up to get an estimate of the volume. Try to measure more often where the radius is changing quickly to reduce the error in your result. If the balloon does not possess rotational symmetry, the basic idea above is the same but the technique is a bit harder. Basically you divide an axis into segments and make an estimate of the cross-sectional area in each segment. Multiply that by the segment width to get an estimate of the volume of that section, and add all these up to get an estimate of the total volume. In either case the accuracy will be improved as the segments get smaller and more numerous, as long as your measurements are sufficiently accurate.

Answer 3

Well, it can be approximated fairly easily. Just take the main shape and consider it a sphere, then use volume of a sphere formula.

Volume = 4/3 X pi X radius^3

To get an exact volume mathematically, you would need to use calculus. But another way to get exact volume, if it is phisically possible, is to see how much water it will hold. Then transfer the water into something easy to measure volume, like a cube, or something.

Answer 4

For a common hot air balloon the easiest approximation is a hemisphere for the top and a truncated cone for the bottom and add the two volumes together.
If you actually have a model, immersing it in water and measuring how much water is displaced is one way.

Answer 5

well, if the balloon components are spherical, then you can easily calculate the internal volume of the sphere(s)...4/3*pi*r^3. by perimeter i assume you mean circumference, so since the circumference of a sphere = 2*pi*r, then r = perimeter/(2*pi) which you can use in the formula for spherical volumes...and you're done.

Answer 6

Cut the volume into parts such that you can calculate each of the volumes. The sum of separated volumes gives you required answer.

Answer 7

Make a scale model of it, throw it in a container full of water and measure how much water is displaced.

Or, make a computer model of it, and calculate volume with software functions.