Hatchet Planimeters

thingiverse

Our school embarked on a three-day journey to Yosemite with the theme of "Restore Hetch Hetchy." Each discipline had to develop a lesson that connected to this theme, and I decided to experiment with an unfamiliar tool to estimate the volume of water stored by the O'Shaughnessy Dam: a planimeter. After researching three types of planimeters – Linear, Polar, and Hatchet – I chose the hatchet planimeter as the most accessible option. With just a dead branch, a patch of dirt, and a map featuring the area to be calculated, students were set. To enhance precision, I aimed to utilize my 3D printer. I introduced Green's theorem and provided an informal proof using Professor Foote's website (http://persweb.wabash.edu/facstaff/footer/Planimeter/HowPlanimetersWork.htm) and another resource (http://user.mendelu.cz/marik/mechmat/planimeters_with_explanations/) to my Algebra 1, Geometry, Algebra 2, Precalculus, and AP Calculus classes. Design 1: My initial idea involved using a pencil in conjunction with a small piece of plastic featuring a tiny tip. The pencil or pen should have a small enough tip to allow the planimeter to rotate freely. However, this design faced several issues: 1) Slight warping of the paper caused large errors. 2) The corners near the hole for the pen/pencil dragged and added additional error. 3) It was challenging to trace exactly over the area due to the small size. This design was suitable because I could print a class set (approximately 45 at a time) efficiently. My printer could have printed up to 100 units if needed. Design 2: Building upon my first design, I created a larger model featuring a hole for a printed "stylus." This allowed students to see the tip of the stylus and trace their area accurately. However, this improvement introduced new issues: 1) If not held correctly, the stylus restricted free movement. 2) The increased size made it more cumbersome to carry on our hiking trip. 3) The tip of the stylus and the cylinder base sometimes struggled to join together seamlessly. I was also concerned that the hatchet tip was a cone rather than a blade or hatchet, which would allow for non-perpendicular movement. Nevertheless, this version remained popular among my students (60% liked it), and I achieved the most accuracy with it. Version 3: My goal was to create a blade or "hatchet" for the end of the stylus, similar to some online examples. I also simplified the connection between the tip of the stylus and the base cylinder. Unfortunately, this version proved less accurate than Version 2 in multiple trials. I attribute this discrepancy to the varying length of the blade when dragged towards or away from the stylus. Unfortunately, time constraints prevented me from developing a fourth version. My primary takeaway from teaching hatchet planimeter techniques was the importance of allowing students (or oneself) ample time to become accustomed to the technique. With each class, I spent two days experimenting with different models and techniques while calibrating the hatchet planimeters. To calibrate, draw a 5cm by 5cm square with a centroid connected to a vertex via a line segment. Measure 10cm away from the center of the square and mark a point. Draw a 2.5cm radius circle around this point. If you can initiate the planimeter at the centroid, follow the line segment to the vertex, trace clockwise, return to the centroid, and end at the edge of the circle, you are half an expert. Completing the same task counterclockwise and ending with the planimeter back at the center of the circle makes you a 100% expert. For further information, please visit this worksheet: https://goo.gl/yZnuAz Feel free to comment or ask questions; I am always eager to challenge myself and learn.