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 two- dimensional 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.