In the Studio: Math for Potters Part 2: Volume Sumi von Dassow

Appears in the Nov/Dec 2021 issue of Pottery Making Illustrated.

In my last article on math for potters (Pottery Making Illustrated May/June 2021), I addressed how to calculate shrinkage from wet clay to its fired, glazed form. Compared to figuring out how wide to make a plate, calculating volume is considerably more complicated. Many potters deal with this issue by trial and error or experience. Other potters deal with it by avoiding it. Volume is often not critical to the things we make as long as they look well-proportioned. A customer rarely worries about exactly how much a mug, a casserole dish, or pitcher holds—if they like the glaze and the size seems about right, that’s usually good enough. However, there are times when a customer wants a mug that holds exactly the amount a coffee maker makes, a fermenting crock that holds a gallon, or even a cremation urn with a specific capacity. How, other than making a bunch of these items and measuring them after they are done, do you figure out volume?

Measuring a Fired Piece

There are several approaches to this question. For the purpose of this article, I’m using a stoneware clay that shrinks 12.5% (typical for mid-range and stoneware clay bodies). If you have a request for a 20-ounce mug, you might measure the mugs you regularly make and see if you have one that holds 20 ounces (1). If you do, measure its height and diameter and apply the formula for linear shrinkage: divide the height and diameter by 7 and multiply by 8. Assuming it is the same shape as the fired mug, it should hold the same amount when finished.

If you have a selection of bisque-fired mugs and you want to know if any of them will hold 20 ounces after shrinkage, fill them with water and measure how much they hold (2). Suppose one holds 25 ounces. How much will it hold after shrinking? If your clay body shrinks a total of 12.5%, typically half of that shrinkage is from wet to dry and half is from bisque to glazed. So your bisque-fired mug is going to shrink 6.25% in the glaze firing. That does not mean it will hold 93.75% as much liquid as it does now! Your mug will shrink in three dimensions—volume will decrease by 93.75% for each dimension. So, the volume will be 25 ounces ×.9375 × .9375 × .9375, which works out to 25 × .824, or 20.6 ounces. Keep in mind that if you want this mug to hold 20 ounces, you can’t fill it to the rim—leave some head space at the top so your customer can lift it to their lips without spilling. (Note: With my cone-6 clay bodies, the shrinkage is about half from wet clay to bone dry, with no measurable shrinkage during a bisque firing, and the other half is in the glaze firing.)

Just for comparison, how much does a pot decrease in volume from wet to glazed? Each dimension of the finished pot will end up at 87.5% (100% minus 12.5%) of the wet dimension, so to calculate the shrinkage in volume, you would multiply the volume of the freshly thrown pot by .875 × .875 × .875. This comes out to .669, which is almost exactly ⅔. No wonder your pots look so much smaller after firing! If you want a crock that will hold 1 gallon of sauerkraut, it has to hold 1.5 gallons when you throw it because ⅔ of 1.5 gallons is 1 gallon. You’ll always need to make your freshly thrown pot hold 1.5 times the volume that you want the finished pot to hold.

Calculating the Volume

However, suppose you have nothing fired, and nothing to measure for reference. How can you calculate how big to throw a pot so it shrinks to the correct volume? The formula for calculating the volume of a cylinder is pi (3.1415) multiplied by the height of the cylinder and by the square of the radius (half the diameter), or
Pi × h × r². 

If your standard mug is pretty close to a cylinder in shape, we can calculate roughly how big to throw it. If you figure your freshly thrown mug will be 4 inches in diameter, that makes the radius 2 inches, and of course the square of 2 is 4. So, we only have to figure out how tall to make it. Knowing that finished mug will shrink to hold ⅔ as much as it would hold wet, you need to make a mug that would hold 30 ounces when freshly thrown. 

There are 231 cubic inches in 1 gallon of water and 128 ounces in a gallon. Dividing the total cubic inches by the number of ounces gives us the volume measurement for 1 ounce of water: 1.8 cubic inches. This means that the volume of 30 ounces is 54 cubic inches. Now we can plug the numbers into the equation to find the height needed. Divide 54 (volume) by 4 (radius squared), and divide again by pi, and the answer is you need to make the mug 4.3 inches tall. Don’t be too hasty—to accurately estimate volume, both the diameter and the height must be measured inside the mug! So your outside diameter will be closer to 4½ inches and the height must be over 4½ inches. Add a ½ inch to the height for the head space so your customer can drink without spilling, and your mug needs to be 5 inches tall. If your mugs are not perfect cylinders, especially if they taper at the foot, you’ll have to make them even taller. 

I used inches and ounces for my example because in the US we still use those measurements. You don’t have to look up numbers like how many cubic inches there are in 1 gallon if you get comfortable with metric measurements such as liters and cubic centimeters, because 1 liter holds exactly 1000 cubic centimeters. In this example, we could size our freshly thrown mug to hold a liter. A liter is just over 1 quart, and 1 quart is 32 ounces, which is about the size we want (before shrinkage). There are 2.54 cm in 1 inch, so if you make your mug about 10 centimeters in diameter that is very close to 4 inches. The radius is 5 cm and the square of the radius is 25. Divide 1000 by 25 and divide the answer (40) by pi, and you find you want to make the inside height 12.7 cm. Round up a bit for that head space, and make it 13 cm high.

There are two types of pots that customers often need measured in volume; one is a fermenting crock (3), and the other is a cremation urn (4). Cremains (the proper word for ashes) measure about 1 cubic inch per pound of body weight. There are 61 cubic inches in a liter, and a standard urn should hold at least three liters so that a 180-pound person’s cremains will fit. You can use either cubic centimeters or cubic inches to calculate how big to make an urn, but for US readers who are comfortable visualizing inches, I’ll use inches. For a standard urn, the wet pot will have to hold a volume of 180 (body weight) × 1.5, or 270 cubic inches. If you made it a perfect cylinder 6 inches in diameter, you would divide 270 by 9 (the square of the radius) and then by pi, for a height of about 9½ inches. Remember, those are inside measurements. If your urn is not a perfect cylinder—if it has a wide shoulder and a narrow foot, you’ll have to factor those variations in, but it will help to imagine a cylinder of that size fitting inside your form. I’m not going to bring in calculus to figure the exact volume! You are an artist, you can do a bit of visualizing. It might even help to find a cylinder about that size to look at while you are throwing—something like a large oatmeal canister. Once you’ve finished and fired your first urn, use a pitcher to fill it with water, measure how many liters it holds, and then you’ll see how close you came to the correct size. 

While an urn is rarely a cylinder, the ideal form for a fermenting crock is almost a perfect cylinder. A crock should be straight sided and just a bit wider at the top than at the foot. So calculating how big to make it if you want it to hold a gallon is simpler. You need to leave some head space in a crock, too—your customer needs to be able to pack the cabbage down in the crock and then put a weight on top of it, so if you were to fill it with water all the way to the rim, it will need to hold 4½ to 5 quarts. Multiply this by 1.5 for the volume of your freshly thrown pot, and you’ll want to aim at about 7–7½ quarts. A quart (i.e. a quarter of a gallon) is 57.75 cubic inches so 7 quarts is 404.25 cubic inches. Let’s divide by pi first this time, and we get about 128. I did that to allow me to quickly calculate different combinations of height and diameter. A diameter of 8 inches would mean dividing 128 by 16 (four squared) for a height of 8 inches. A diameter of 7 inches would require a height of around 10 inches. Something around that size sounds pretty reasonable for a crock, actually, keeping in mind that these are the inside dimensions. In this case, if you make the sides perfectly straight and wider at the top than at the foot, you’ll measure the diameter half-way up.

I hope all this math doesn’t scare you off from trying to make pots that hold a specific volume. Use it as a fun exercise—play around with guessing how much your pot will hold after it is fired, even if nobody is asking, and eventually when a customer does want a specific volume, you’ll be ready to fill the order. 

Sumi von Dassow is the author of In the Potter’s Kitchen, available at http://ceramicartsnetwork.org/store/in-the-potters-kitchen.