I’d like to thank Matt Parker for introducing me to diameter tapes (or D-tapes). These are measuring tapes used by foresters to measure the diameters of trees. The forester wraps the measuring tape around a tree as if measuring the circumference, but the scale on the tape is adjusted so that the measurement gives the diameter (assuming the tree has a circular cross section). It uses the fact that C=πD.
Parker mentioned the existence of these measuring tapes in his talk at Gathering 4 Gardner 12 and he gave every attendee a D-tape that he made (see below).
That got me thinking. If you know the circumference, you can compute the diameter. But you can also compute the area. Also, if you wrap the tape around the equator of a sphere, you can compute the the volume and the surface area of the sphere.
Thus, inspired by this, I made measuring tapes to measure the diameter and area of a circle and the volume and surface area of a sphere. The top of each tape is marked off in centimeters, so it measures the circumference. The bottoms can be used to measure the diameter of the circle, the cross-sectional area, the volume of the sphere, and the circumference of the sphere.
I’ve also created a printable pdf of these measuring tapes, which should, I hope, print to the right scale so that 1 cm on the measuring tape is actually 1 cm. (Note that although a 25 cm ruler looks long, when you wrap it around a circle, you find that the diameter is only 8 cm. Not very big.)
It was a fun exercise to figure out how to mark and make the rulers. Except for the diameter ruler, the relationships aren’t linear. (I set everything up in Excel, then I moved the info over to a Geogebra spreadsheet. I used GeoGebra to draw everything. I exported the images as pdfs. Then I cleaned them up and added the text in Adobe Illustrator.)