How do you calculate the volume on an object with an irregular shape?

Question

- I need to know how to calculate the volume of a ballon for a science fair project. I know it involves calculus. Could someone explain in detail how you do it, and or give me a formula?

Answer 1

This is not a straightforward question.

If you assume that the balloon is a perfect sphere, or a close approximation then you could use the calculation for volume of a sphere: 4/3 pi x r^2

However, few balloons are perfect spheres. I would go a different route and submerge the balloon in liquid to measure displacement.

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

If the shape is truly irregular, you cannot calculate the volume. The fact that you can calculate mean there is some sort of regularity in the shape, and you can arrive at a mathematical model to describe the shape.

You can perhaps use an approximation method by drawing the shape and breaking it into smaller cubes. Or - you can fill the balloon with liquid and place it in more regular container. Or - you can put the balloon in a tank and measure the water displacement.

Answer 4

If you have a measuring glass, fill it to a certain level that will cover the object, e.g. to 100 ml. Put the object into the water then read its new level. If the level is 138 ml, it means that the object volume is 138-100= 38 ml.

You can also try by putting on a large dish a glass or a can which is large enough to for the object to be put in. Fill the glass or can with water or aquadest until full. Put into it the object slowly and carefully to let the water spill over into the dish. Weigh the water on the dish, it will show its volume. If the water is 29.5 grams, it means that the volume of the object is 29.5 ml.

Answer 5

You do need calculus.
You have to figure out how much the radius of the circle changes.
You need to "divide" the balloon into really thing cylinders, the thinner they are, the more precise your volume will be.
You can find the radii in different areas by measuring the circumference and dividing by pi. Then add up all the volumes of the cylinders. You can use Siemman's sum for that.

Answer 6

isnt balloon a perfect sphere? u can try 4/3 multiplied by 3 .14 and then by cube of the radius
or try the archimedes method . immerse it in a beaker of water fiiled to the brim collect the overflow in a measuring jar. the amt of overflow gives the volume. but since the balloon is light , a sinker ( cork) may have to b tied to the balloon so that it gts completely immersed

Answer 7

Use Archimedes' bathtub method. Fill a container with water, stick the balloon under water and measure how much spills out. (Or you can calibrate the container first with know volumes of water and then just measure how much the level rises once the balloon is totally under water.

Answer 8

Assuming you mean "of" an object:
You don't calculate it if it is irregular.
(Irregular means you do not have a formula for it.)
You use a overflow tank and measure the displaced liquid.
(Watch out, you may then get an urge to run naked down the street yelling Eureka.)