Centroids without Calculus

  • Objective
  • Level
  • Prerequisites
  • Platform
  • Instructor's Notes
  • Credits
  •       Find my centroid!
     
    Objective: To demonstrate a simple physical method of determining the centroid of an irregular region.

    Level: Precalculus or Integral Calculus 

    Prerequisites: None or integral methods for determining the centroid of a planar region.

    Platform: A classroom and about 20 minutes.

    Instructor's Notes:

    Centroid of an Irregular Region

    After we have discussed (and derived formulas for and worked problems involving) the centroid of a planar region bounded between two curves in our calculus, I ask my students how they would determine the centroid of the state of Washington  -- “Where would Washington balance if was uniformly thick and dense?"  Since many of my calculus students plan to major in engineering and are “hands on" types, someone usually suggests cutting out a map of the state and actually trying to balance it.  At that point I agree that is a fine idea, and I produce the state of “chaos" (see Figure 1) and ask someone in the front row to find the centroid by trying to balance it on a finger.  (Sometimes I hand a rectangular piece of cardboard and scissors to someone and ask them to create a “crazy" shape for me.) 

                                          Figure 1.

    Then I pull out a pin and a piece of string, and pin the shape (somewhere along its edge) to a classroom bulletin board (using a loose enough hole so the shape can swing on the pin). I explain that the centroid (center of mass) of the region is attracted by gravity and ends up on the line connecting the pin with the center of the earth and that this is the same line segment created by a weight hanging by string from the pin (see Figure 2).  So we know that the centroid of the region lies on the line created by the hanging string. 
                                                           Figure 2.

    If we repeat hanging the region from another point near its edge, we get another line that must contain the centroid of the region.  Thus the centroid of our region must lie on both lines, so it lies at their intersection, and we are done.  (See Figure 3.)

                                                                 Figure 3.

    As a check, I always do this a third time to get a third line, and, if everything was working correctly (and swinging freely), the three lines intersect at a point or very close to a point (see Figure 4).  Then I show that we actually found the centroid by showing that I can balance the region on my finger at the centroid.
     

                                              Figure 4.

    After this demonstration, I usually give them an assignment (to be turned in a couple days) to find the centroid of a city, state or country that interests them (with the restriction that no one can be interested in a rectangular state such as Nevada).  Since I have a lot of foreign students, I get to see the centroids of many countries -- did you know the centroid of Brazil is very close to the capital city Brasilia, and that was part of the reason the location was chosen to be the capital?

    Center of Population:

    After the “uniform thickness, uniform density" case, I ask how we could find the location of the “center of population" of our state, and someone usually suggests attaching weights proportional to the populations of the major cities at the locations of the cities.  Sometimes we simply discuss that idea and sometimes I have some fishing weights that I attach to our crazy shape (see Figure 5).  Then I quickly take the “weighted region" and determine a couple lines using the string to find the 'balance point' for a populated region. (The shiny squares in Figure 5 are clear mounting squares which make it easy to adjoin weights to the figure.)
     

                                                           Figure 5.

    One other situation:

    If there are only a few minutes left in class, I demonstrate that the centroid of a region does not have to actually be in the region (Figures 6 and 7).

                           Figure 6.                                           Figure 7.

    Materials:

    a piece of rigid cardboard or foamboard
    a piece of string
    a pin
    something to act as a weight on the string

    Comments:

    This is a simple and quick demonstration that seems to appeal to the “hands on" students in the class.  And they seem to appreciate that I am willing to do something “real" for them after all of the derivations and calculations that we had been doing to find centroids of regions determined by formulas in our calculus class.  Even some of the more “theory" oriented students seem to have a bit of fun with the assignment of finding the centroid of a region of interest to them. 

    I require the students to turn in a paragraph telling me what region they chose, why they chose that region, and describing the method they used to determine the centroid.  I bring a box to class on the day the assignments are due because they also have to turn in their materials.  (I look for pin holes on their regions to see that they actually did the experiment.)

    Sometimes as part of the same assignment I have them use the “official math department teeter-totter" to find their own center of mass.

    This hands on activity can also be used in a precalculus course or as part of a geometry course for education majors at all levels.

    Other resources:

    Centroid of a triangle.

    The following link lets you drag to change the vertices of a triangle and shows that the centroid remains at the intersection of the medians.

    http://www.keypress.com/sketchpad/javasketchpad/gallery/pages/centroid.php

    'This is an interesting link where students can see an illustration of how a centroid of a triangle is calculated. Instructor or student can click on any of t he three vertices and change the shape of the triangle as desired. A good visual aid for students to understand theory and math of centroids, pictorially.' (by Un Jung Sin, student at Temple University, Spring 2001) 

    The next link briefly discuss the centroid of a tetrahedron in three space.
    http://www.pballew.net/centroid.html


    A table of centroids.

    A table of centroid locations for a few common line segments, plane regions, and solids. Includes formulas and pictures.
    http://www.engineering.com/content/ContentDisplay?contentId=41005016


    Credits:  This demo was submitted by 

    Dale Hoffman
    Department of Mathematics 
    Bellevue Community College

    and is included in Demos with Positive Impact with his permission.

    We also appreciate the work of Un Jung Sin in reviewing the web sites mentioned above and recommending that several be included as other resources.


    DRH 3/23/01   Last Updated 5/18/2006

    Since 3/1/2002