How To Work Out Cubic Conversions
More recently, we've received a lot of questions relating to cubic conversions. Whether you're attempting to calculate the cubic volume of your fridge freezer or trying to decide how many cubic feet of topsoil you might need for your garden, the calculation query has the same principle.
Yesterday, we received this question:
"Trying to figure out the volume of this chest freezer. Here are the dimensions: 22" wide, 29 1/2" high, and 25 3/4" depth. Can you please tell me the volume in cubic feet and cubic meters???"
Well, it’s hardly surprising if that little problem induces a dull pain between your eyes. We have at least four levels of complexity there, each nested inside the next like a Russian Matroshka doll.
Working outwards from the core, we find:
- Working with fractions
- Converting lengths into cubic measurements
- Multiplying Imperial measurements
- Converting Imperial into Metric
If we unpack those one at a time, the correct answer should appear. But first, we need to consider which way round to approach the problem:
- Work out the volume in cubic feet and then convert into cubic meters.
- Convert the dimensions into centimeters, convert those into Metric volume and convert back to cubic feet.
I’ll do it both ways because the first method gives more insight into how to work with Imperial measurements. Then I’ll show the second way of working the problem, so you can see how much easier it is!
Thinking in fractions
I have two advantages when it comes to working in fractions of inches. First, I’m a Brit who can remember the days when inches, feet and yards were all we had. Twelve inches to the foot, three feet to the yard, 1,760 yards to the mile. Simple!
The second advantage is that my Dad was a carpenter who often ‘persuaded’ me to help out with his latest project in the shed he used as his workshop. In carpentry, you work down to 1/64 of an inch, so you often have to do sums like “Make a 2 9/32” x 1 5/64” mortise cutout exactly in the center of a batten that measures 15 1/2 x 3 ¾” Well, I won’t bore you with that particular series of sums but the point is that you soon learn to reduce fractions to lowest common denominators to avoid going mad.
So let’s apply the same approach to the chest freezer, which you’ll remember was:
W 22" x D 25 3/4" x H 29 1/2"Now the smallest unit involved there is one quarter of an inch, so we’ll re-express those dimensions in quarter-inch units, multiplying all the whole inches by four and adding the fractions expressed in quarters.
- W (22 x 4) = 88
- D (25x4) +3 = 103
- H (29x4) +2 = 118
So now we multiply the width by the depth (88 x 103) to find the floor area of the unit, which comes out at 9,064 ‘square quarter inches’ and then multiply that by the height (9,064 x 118) to find that there the volume of our chest freezer is made up of exactly 1,069,552 quarter-inch sized cubes.
Working with cubes
Wow, that’s an awful lot of cubes. How can we turn that into something sensible?
Imagine we were working in cubic half inches, the most basic fraction. The illustration above shows what a cubic inch looks like divided into half-inch cubes, two along the bottom, two along the side and a second layer on top. So you can see there are four cubes in each layer, making eight units in all (2 x 2 x 2 = 8).
But we’re working in quarter inch cubes, four to the inch. This (4 x 4 x 4 = 64) means there are sixty-four quarter inch cubes in each cubic inch. Dividing 1,069,552 by 64 gives us 16,711.75 cubic inches.
Multiplying Imperial measurements
Now, if this had been a Metric sum and those were cubic centimeters, the rest would be easy, just a question of moving the decimal point over to the left. But they’re not, it isn’t and so we can’t. Instead, we have to remember there are 12 inches to the foot, which means there are a hefty (12 x 12 x 12) = 1,728 cubic inches in a cubic foot. So, more division: 16,711.75 ÷1,728 = 9.67115162 cubic feet. Rounded down to two decimal places, that makes the first answer 9.67 cubic feet. And if you wanted to express that in cubic yards, given that there are 27 cubic feet (3 x 3 x 3) in a cubic yard, the answer would be around 0.358.
Why on Earth would anyone want to know the answer in cubic yards? Well, builder’s sand comes in cubic yards so if you wanted to fill the chest freezer with that...
Incidentally, a lot of folks are surprised at how big a cubic yard actually is. Twenty-seven cubic feet or 46,656 cubic inches equals a heck of a lot of sand – enough to spread a half-inch thick layer over an area of 250 square feet!
Anyway, there’s not a lot of difference between a cubic yard and a cubic meter, so it looks like we’re getting close to the second part of our answer. But getting to a precise number is going to need a little bit more calculator punching.
Imperial to Metric
There are two ways to do this, the internet way and the old-fashioned way. The new way is to use one of those widgets you find online that converts Imperial to Metric; in this case cubic feet to cubic meters. Well, luckily we kept our precise measurement of 9.67115162 cubic feet, which, hey presto, we find equals 0.2738565m3, or more sensibly, 0.274 cubic meters.
Without ‘cheating’, the precise way to do this is to know that one foot equals 0.3048 of a meter and that therefore 0.3048 cubed should turn one cubic foot into 0.028316847 of a cubic meter. So that’s our conversion factor; multiplying that by our 9.67115162 cubic feet should give us 0.274. Let’s see….click, click, click…. 0.2738565. Hey, it works!
How did we know to multiply rather than divide? Common sense, really. Obviously the cubic meters figure is going to be smaller than the cubic feet one, and since the conversion factor is less than a whole number, it follows that as a multiplier it will make your original number smaller as well. There’s another way to make the conversion, less precise but slightly easier to do with pencil and paper. It’s possible to remember that 3.28 feet make a meter and so there are 35.29 (3.28 x 3.28 x 3.28) cubic feet in a cubic meter. Therefore each cubic foot will be just over one 35th of a cubic meter so 9.67 cubic feet divided by 35 equals 0.276, which is pretty close.
I did promise to show you the other way of approaching this challenge and I always keep my promises, so...￼
Getting out of the (imperial) box
The approach here is what any sensible person would advise, get it into the Metric domain as soon as you can – and stay there as long as possible!
Let’s look at those original chest freezer measurements again...
W 22" x D 25 3/4" x H 29 1/2"
Have you spotted what I just did? There are no nasty eighths, sixteenths or thirty-seconds in there, so each one of those measurements can be expressed as a nice neat decimal:
W 22.0" x D 25.75" x H 29.5"
Now we’re halfway into the easier world of Metric; all we need is to turn those inches into something easier. Remember this if you remember nothing else:
One inch equals 2.54 centimeters.
1” = 2.54 cm. Have you got that? Good, now the rest is easy.
Turning the dimensions into centimeters by multiplying each one by 2.54 gives us:W 55.88cm x D = 65.405 x H 74.93
Now we just do as we did before: multiply 55.88 cm width x 65.405 cm depth to find that the floor area of the freezer is 3,654.8314 square centimeters. Multiplying area by the height of 74.93cm gives volume of 273,856.517 cubic centimeters.
Since there are one million (100 x 100 x 100) cubic centimeters in a cubic meter, we just need to move the decimal point six places to the left to turn 273,856.5 cc into 0.2738565. Job done… nearly.
We need to know cubic feet as well, remember; so to convert those cubic meters, we need to go back to our conversion 0.028316847. We used this as a multiplier when going down from cubic feet to cubic meters so presumably we need to divide by the conversion factor when going the other way: 0.2738565 divided by 0.028316847 equals… well, whaddyaknow… 9.6711509 cubic feet. QED.One last note: the second way is definitely the best. Working through it showed me where I’d made an error with the original inch fractions during the first method. Keep it Metric, people.
Written by Nick Valentine
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Last update: 12 October 2013
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