What’s the balance point?
By: Victoria Lee
Introduction
My assignment was to investigate which Method - Ruler,
Construction, and Plumb Bob - would be best for an
artist to use to find the centroid of his/her shape. I tested
each of the three methods on a triangle to determine
their comparative accuracy. I also tested the Plumb Bob
method on an irregular 6-sided polygon. A description of
my work, findings, and conclusion follow in this
presentation.
Centroid Theorem
The centroid is also known as the balance point. A twodimensional object can be balanced at a single point. For
a triangle, the centroid, or balance point, is the
intersection of the medians. Each median is the line
segment from the midpoint of a side to its opposite
vertex. In addition to being the balance point, the
centroid marks the point from which each median is
divided - ⅔ from the centroid to the vertex, and ⅓ from
the midpoint to the centroid.
Centroid Theorem
Triangle - Method 1: Ruler
For finding the centroid with a ruler, I measured each side
of the triangle with a ruler and found the midpoint which
was equidistant from each adjacent vertex. Then I drew
the median, which is the line from the midpoint to the
opposite vertex. I drew all three medians, and where
they intersected is the centroid, which I labeled “C1”.
Triangle - Method 2: Construction
For finding the centroid by doing a construction, I had to
use a homemade compass because the triangle was too
big for a standard-sized compass. I made a compass
using string, a push pin and a pencil. With the compass,
I drew arcs intersecting each side of the triangle to
determine the midpoint. This method caused several
difficulties. First, the push pin could not be inserted
exactly on the vertex. Second, the pencil doesn’t keep
it’s angle to the string. Third, tracing arcs that were the
right size and that intersected properly was difficult.
Triangle - Method 2: Construction
Despite the difficulties, I managed to find each midpoint
from which I drew the median connecting it to the opposite
vertex. The intersection of the three medians is the
centroid which I labeled “C2” and was the same centroid
as I found using the Ruler Method - C1.
Triangle - Method 3: Plumb Bob
For finding the centroid using the Plumb Bob method, I
had to dangle a piece of weighted string from two
vertices and mark the plumb lines. The intersection of
the plumb lines is the centroid, “C3”, which was the
same as C1 and C2, the centroids found using the other
two methods. This method is easy in the sense of being
quick, but was difficult because it was hard to trace the
line accurately while holding the weighted string.
Polygon - Plumb Bob Method
For finding the centroid of the polygon using the Plumb
Bob method, I dangled a weighted string from two
vertices and then traced the plumb lines. After tracing
the lines with a straight edge, I marked the point of
intersection, otherwise known as the centroid. This
method was easy and quick, but was somewhat
frustrating when trying to accurately mark the plumb
lines while suspending the polygon from a vertex.
Triangle & Polygon - Summary
For the triangle, all of the methods for finding the centroid
concurred. In other words, each method found the same
point which is the centroid, or the balance point. The
centroid theorem is proven because it found the balance
point for the triangle. Of all three methods, I would
suggest the Ruler Method because it was easy and you
would be working with solid numbers and measurements
instead of homemade tools. Fortunately, since we only
used the Plumb Bob method for the polygon, I found it
just as accurate as the Ruler Method, although a little bit
more difficult to use.
Conclusion
Dear Jessica (artist), all three methods were equally
accurate in finding the centroid of the given shape.
However, I would suggest that you use the Ruler Method
because it involves numerical values while finding the
centroid. If you are unable to use the Ruler Method due
to the shape of the artwork, the Plumb Bob method will
surely find the centroid so that you will be able to display
your freestanding artwork using the balance point, the
centroid.