# Q & A: Density of an Irregular Shape

Q:
How do you find the Density equation of the an Irregular Shape?
- Anonymous
A:
 The density of something is just the mass divided by the volume: D = m/V The mass you can measure on a balance or a scale, and the volume is the amount of space the object occupies. You can find the volume of an irregular object by immersing it in water in a beaker or other container with volume markings, and by seeing how much the level goes up. You can use this relation backwards too -- if you know what material an object is made of, you can look up its density in a book. Measuring its mass allows you to compute the volume. Tom

(published on 10/22/2007)

## Follow-Up #1: Another way of measuring density

Q:
How to find density of an object of irrigular shape by using weight in air and weight in water method.
- Abdullahi Ahmed Abdi (age 50 years)
Hargeisa,Somaliland
A:
As described in the previous question the density is given by D = M/V.  If you don't know V then you have to compare the density of the unknown to that of a known density, e.g. water.
Call the density of the object Do and the density of water Dwater.   The weight of the object in air is then Mair = Do * V.   The weight of the object in water is Mwater = (Do - Dwater) * V.
Dividing one equation by the other the volume V cancels out and the ratio of the two measurements is  Mair/Mwater =  Do/(Do - Dwater).   Using the value of  Dwater as 1 gram per cubic centimeter you can easily solve this equation for the density of the object Do.

LeeH

(published on 10/22/2007)

## Follow-Up #2: density simplified

Q:
How do you find the density of an irregular shape ? (please make it easy to understand im not really 25 im much much younger im in 6th grade and u did have some stuff about the density of an irregular shape stuff but it was really confusing and i have to put my stuff in my own words for science homework and ur stuff cant really be put in your own words when you put in answers make sure all ages are able to understand it)
- sushie (age 25)
America
A:
The density just means how much mass per volume. That means mass (M) divided by volume (V).

It's easy to find the mass M. You can just weigh something on a scale. So the trick is to find the volume V of the irregularly shaped object.  As Tom wrote earlier, there's a way to do this.

Take some water,  enough to submerge your object. Mark the level of the water. As the object goes into the water, it will force the water up in the container. Mark the level of the water when the object is completely under.

Now take the object out. You can pour water in from a measuring cup (say one marked in liters) to see how much is needed to fill from the lower mark to the upper mark. That volume is the same as the object's volume.

Mike W.

According to legend, Archimedes figured all this out in the third century B.C. when the Greek king asked him to verify that his gold crown was really gold and not some gold plated fake.  While taking a bath Archimedes noticed the water level rising when he got into the tub.  He got the idea of how to measure density and, again according to legend, was so elated with his discovery he ran naked down the street shouting 'Eureka, I have found it'.  Mike's explanation is an example of what is now called 'Archimedes Principle'.

LeeH

(published on 11/17/2010)

## Follow-Up #3: How to find the density of an irregular object?

Q:
How can I find the density of an irregular object using improvised materials
- DANIEL (age 18)
ASHANTI,GHANA
A:
What a fun question! The Greek mathematician Archimedes is said to have faced a similar problem when he was asked by King Hiero II to determine whether or not the king's new crown had been made with pure gold, as requested. Of course, the easiest course of action would be to melt the crown down to a less irregular shape and compare its mass to its volume, but Archimedes couldn't do that with something as valuable as a crown! He considered the problem at length and, while in the bath one night, Archimedes noticed the change in the water level in the tub as he got in, and he had an idea. The crown, once fully submerged, would displace an amount of water equal to its own volume, so he could calculate its density by dividing its mass by the volume of the water displaced and compare that density to pure gold.

If you don't mind, I'm going to assume for the sake of simplicity that we're discussing an object small enough for measurement on a household scale. In order to determine the density of your object, you will need the following:

-a scale (the more precise, the better)
-any sort of device for volumetric measurement (e.g. measuring cups, a liter bottle or empty gallon container)
-a large container (I recommend a bucket or a mixing bowl... something large enough to hold your object and a significant quantity of liquid)

First you need to find the mass of your object. If it is small enough, I recommend a postage scale, or a trade scale (the kind with which you'd weigh fruits or vegetables at the grocery store). If you have access to either of these, finding the object's mass should be as easy as placing it on the scale and choosing convenient units. Grams or kilograms would be best.

If you have only a bathroom scale, you can first weigh yourself, then weigh yourself holding the object. Try to be as accurate as possible with your measurements, and then subtract the former from the latter to find the mass of the object. This process is commonly referred to as "taring."

Now we need to find the volume of the object. If you're lucky enough to have a container with volumes marked on it, you can simply fill said container with just enough water to submerge the object. Measure the volume of the water without the object in it, then carefully and completely submerge the object and note the new volume of the liquid, taking care not to throw your data off with the addition of any foreign objects to the system (for instance, don't use your hand to push the object down and then measure the volume with your fingers in the water). Once more, tare the two values -- subtract the former from the latter -- to find the volume of your object.

If you don't have a marked container, the procedure will be a little bit different. Submerge the object in the water (this time, the initial volume of the water in the container is irrelevant) following the same guidelines outlined above, then mark the volume the water reaches and remove the object. Use your smaller device to measure out the volume of water it takes to exactly meet that mark. The volume of water you use will be equal to the volume of your object.

Assuming you've worked carefully through the above instructions, finding a relatively accurate approximation of the object's density should be a piece of cake from here. The average density ρ will be given by the expression ρ = m/V, where m denotes its mass and V its volume. You may now divide your measurements accordingly to find this density. Depending upon the units of the household scale/container you used, it may be necessary to do some conversion to get your object's density in the standard, meaningful units of kilograms per cubic meter, kg/m3. Here are some helpful conversion tables...

Remember that this density is probably not uniform. Most objects will have internal regions of greater and lesser density -- what you've computed is just an average.

Hope that helps!

-Becca

(published on 03/03/2011)