Champaign-Urbana Herb Society

Hypertufa Math by Dianna Visek

Our recent hypertufa workshop printed out the importance of being able to calculate how much hypertufa is needed for any particular project. Anna Bergvelt kindly prepared two bags of mix for each participant. I used my big bag during the workshop and took my small bag home. Now the question: How big a bowl can I make with my small bag of mix?

I measured the mix, did some calculations on my mixing bowls, selected one and started covering it. I didn't have quite enough! So I turned to The VNR Concise Encyclopedia of Mathematics to find out how I had goofed. To enable us to calculate, we will assume that all bowls are portions of a sphere. Mathematicians call a bowl a ”spherical cap” and a bowl-shaped solid a ”spherical segment.” Hypertufa experts recommend that the walls be at least 1.5 inches thick. We need enough hypertufa to fill the space between the bowl we use as a mold and an imaginary bowl. If we put the hypertufa on the inside of the mold, the radius of the imaginary bowl will be 1.5 inches smaller. If we put the hypertufa on the outside of the mold, the radius of the imaginary bowl will be 1.5 inches bigger. If you want thicker walls, then adjust accordingly.

Measure the bowl you’re thinking of using. You need the height, which we call H, and the diameter. Divide the diameter by 2 to get the radius, which we call R. If R is significantly less than H, you may want to treat your bowl as a cylinder and use the formula given later in this article. If R is greater than or equal to H, then your bowl is more spherical. The equation for the volume of a spherical segment is V=pH(3R²+H²)/6. For p you can use 3.14 or 3.14159. You need to calculate the volume of your real bowl and the volume of the imaginary bowl, then subtract to get the volume of the space in between them. If you’ve measured in inches, the result will be in cubic inches. We now need to convert cubic inches into cups and other common units of measurement. I calculated 12.96 cubic inches per cup. So divide your volume in cubic inches by 12.96 to get the number of cups you need. The fact that there are 4 cups in a quart and 4 quarts in a gallon will help you convert to larger units if you wish.

The hypertufa recipe we used was 1 part Portland cement, 1.5 parts perlite and 1.5 parts peat moss. (If a large coffee can is used to measure a part, then a handful of concrete reinforcing fibers should be added to the mix.) On page 23 of Making Concrete Garden Ornaments, Sherri Warner Hunter says that Portland cement adds very little to the volume of concrete since its fine particles mixed in water fit between the particles of aggregate. Therefore, she recommends using only the aggregate to measure or calculate volume.

Since our aggregate is equal parts perlite and peat moss, we divide our volume in cups by 2 to get the amount of perlite and the amount of peat we need. Since that amount represents 1.5 parts in the recipe above, we multiply by 2/3 to get 1 part, which is the amount of Portland cement we need. We’re done calculating! To be safe, we might want to make a bit extra, maybe 5%.

As mentioned, bowls whose radii are significantly less than their height may be closer to a cylinder than a spherical cap. In that case, measure height H, the radius of the base RB and the radius RA of an average part of the bowl. If it's too hard to get inside the cylinder to measure the radius, you can use a tape measure to measure the circumference. The radius will be RA=C/(2p).

We’re going to calculate the volume of the base and the volume of the walls separately and then add them together. Since the area of a circle is A=pR2, the volume of hypertufa needed for the base is VB=pRB2T, where T is the desired thickness. If the hypertufa will be put inside the cylinder, then the volume needed for the walls is VCin= p(H-T)(RA2-(RA-T)2). If the hypertufa will be put outsid the cylinder, then the volume needed for the walls is VCout=p(H+T)((RA+T)2-RA2). Add the volume of the base to the volume of the walls and you’re done.

Marcia Eischen finds that old-fashioned oval roasting pans make good molds for hypertufa troughs. For ease of calculation, let's assume a roasting pan is an ellipse with straight sides. If the width is W, the length is L and the height is H, then the equation for the volume of the roasting pan is V=pHLW/4. Since we’re dealing with widths and lengths, rather than radii, L and W for the imaginary roasting pan will be 3 inches smaller or larger than the mold. As before with the spherical segments, we need to subtract the smaller volume from the bigger one to calculate the amount of hypertufa needed. If you want to make a rectangular trough, the equation for the volume of a rectangle is V=LWH.

If you’re going to be filling a mold, rather than lining or covering it, the easiest way to figure out quantities is to fill it with perlite and measure the amount. As above, divide by 2 to get the quantity of perlite and peat moss. Then multiply by 2/3 to get the amount of Portland cement. If you don’t like calculating, you can guesstimate how much you need and make more if you run out.

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