Section 19–2 Locating the center of mass

(Pappus’ area theorem / Pappus’ volume theorem / Centroid of a triangle)

 

In this section, Feynman explains Pappus’ centroid theorems—for surface area and volume—and illustrates the idea with the centroid of a triangle. While the triangle example serves as a simple case, its core idea includes the theorems, which are powerful geometric shortcuts in problem solving. For that reason, the title ‘Locating the Center of Mass’ seems too narrow. A more accurate and appropriate title could be ‘Pappus’s Centroid Theorems.’

 

1. Pappus’ area theorem:

“Then it turns out that the area which is swept by a plane curved line, when it moves as before, is the distance that the center of mass moves times the length of the line (Feynman et al., 1963, p. 19–4).”

 

Pappus' area theorem describes a method for calculating the area of a surface or length of a curve. In simple terms, it states: The surface area generated by rotating a planar curve about an external axis is equal to the product of the curve’s length and the distance traveled by its centroid (center of mass). Formally, if a planar curve of length L is rotated about the external axis that does not intersect the curve, the surface area A generated is: A=L´d where d is the path length traced by the centroid. This is a useful theorem because it bypasses integration: once the curve’s length and centroid path are known, the surface area follows immediately. In short: Pappus reduces a potentially messy calculation to the simple rule: Surface area = curve length × centroid path.

Source: Wikipedia

Necessary Conditions:

The Pappus centroid theorems—both for surface area and for volume—are elegantly simple, but they hold only under specific conditions:

1.      Planar curve or area – The figure to be rotated must lie entirely in a plane. If it is non-planar, such as a helix or the surface of a sphere, the resulting rotation can produce a self-overlapping, ill-defined surface or solid.

2.      External axis – The axis of rotation must lie in the same plane as the figure. If the axis is tilted out of the plane, the centroid traces a skewed three-dimensional path (e.g., elliptical path) rather than a simple circle.

3.      Non-crossing axis – The axis must remain outside the region’s interior. If it passes through the interior, the resulting surface or solid intersects itself, and the formula no longer applies.

These conditions are essential: if any are violated, Pappus’ theorems cease to hold. Some mathematicians might suggest that Feynman could have emphasized these limitations more explicitly in his presentation of the theorems.

 

2. Pappus’ volume theorem:

“…if we take any closed area in a plane and generate a solid by moving it through space such that each point is always moved perpendicular to the plane of the area, the resulting solid has a total volume equal to the area of the cross section times the distance that the center of mass moved! (Feynman et al., 1963, p. 19–4).”

 

Pappus' volume theorem gives a direct way to calculate the volume of a solid or area of a surface. In simple terms, it states: The volume of a solid generated by rotating a planar region about an external axis is equal to the product of the region’s area and the distance traveled by its centroid. Formally, if a planar region of area A is rotated about an external axis that does not intersect the region, the volume V of the solid generated is: V = A ´ d where d is the path length traced by the centroid during the rotation. This theorem is useful because it bypasses integration: once the area and centroid are known, the volume follows immediately. In short: Pappus reduces a potentially messy calculation to the simple rule: Volume = planar area × centroid path.

 

“There is another theorem of Pappus which is a special case of the above one, and therefore equally true…... (The line can be thought of as a very narrow area, and the previous theorem can be applied to it.) (Feynman et al., 1963, p. 19–4).”

 

Pappus Centroid Theorem

Some mathematicians may have a different view from Feynman as explained below: Pappus’ two centroid theorems—one for surface area and one for volume—are in fact equivalent and can be seen as two aspects of the same principle. The volume theorem can be derived from the area theorem by viewing a region as a collection of infinitesimal line segments, each generating a strip of surface area whose integration yields the total volume. Conversely, the area theorem can be derived from the volume theorem by treating a curve as the limiting case of an infinitesimally thin strip of area, whose rotated volume reduces in the limit to the surface area generated by the curve. In this sense, each theorem implicitly contains the other, and together they express a single unifying principle: the quantity being rotated—whether length or area—multiplied by the path of its centroid yields the resulting surface or volume. Hence, it is natural to group them under the name Pappus’ Centroid Theorem, with the outcome determined by whether the rotation involves a curve or a region.

 

3. Centroid of a triangle:

“… if we wish to find the center of mass of a right triangle of base D and height H (Fig. 19–2), we might solve the problem in the following way (Feynman et al., 1963, p. 19–4).”

 

Pappus’ volume theorem can be easily applied to a right triangle with base D and height H. By rotating the triangle 360 degrees about an axis through H (Fig. 19–2), it generates a cone. The volume of this cone, with height H and base radius D is πD²H/3, which is exactly one third the volume of the smallest cylinder that can enclose it. Based on Pappus’ theorem, the cone’s volume (πD²H/3) is equal to the area of the triangle (½HD) times the circumference (2πX) moved by the centroid. Thus, (2πX)(½HD) = πD²H/3, and we can obtain the x-component of the centroid as X = D/3. Similarly, rotating the right triangle about the other axis, we can deduce the y-component of the centroid as Y = H/3.

 

Physical Interpretation

Pappus’ centroid theorem can be understood physically using the idea of mass. Imagine a planar figure rotating about an external axis: each tiny mass element moves along a circular path, contributing a small segment proportional to its distance from the axis. The center of mass (centroid), by definition, is the single “average” position weighted by all these elements. This means we can think of the total motion as if one mass—the sum of all the elements—moves along a circle whose radius is the centroid’s distance from the axis. From this perspective, the surface area or volume generated by the rotation is simply the length or area of the figure multiplied by the distance traveled by the centroid. In short, Pappus’ theorem arises naturally from the principle that the motion of a distributed mass can be represented by the motion of its center of mass, transforming a complicated integral into a neat geometric shortcut.

 

Applying Pappus’ theorem via Work 

Interestingly, Pappus’ theorem can be illustrated using the concept of work (Levi, 2009). When a planar shape rotates around an axis to generate a solid of revolution, imagine filling the volume with a fluid and slowly compressing it with a piston. The infinitesimal work done on a thin disk of thickness dx is dW=P dV =PA dx, where A is the area of the disk. Summing over all disks, the total work done corresponds to the area of the shape multiplied by the distance traveled by its centroid—exactly the volume generated by the rotation. This approach essentially treats the rotational motion as a series of infinitesimal “pushing” operations, showing that the total volume equals the product of the shape’s area and the centroid’s path — a physical demonstration of Pappus’ theorem (see figure below).

Source: (Levi, 2009).

Review Questions:

1. How would you state Pappus’ centroid theorem for surface area?

2. How would you state Pappus’ centroid theorem for volume?

3. How would you locate the centroid of a right triangle?

 

Historical note:

Pappus of Alexandria (4th century AD) first formulated the centroid theorems in his Collection. His reasoning was expressed in the language of Greek geometry, relying on the concept of proportions and the center of gravity rather than integral calculus. By showing that the centroid of a figure traces a circular path under rotation, and relating that motion to the resulting area or volume, he provided a geometrically insightful formulation, though not a step-by-step proof in the modern sense.

In the 17th century, Paul Guldin republished these results in his Centrobaryca (1640–41), embedding them within a systematic theory of centers of gravity. Drawing on Archimedean principles—particularly the concepts of balance and moments—he offered a deductive justification of the theorems. Although Guldin lacked the tools of calculus, his treatment was more rigorous than Pappus’, because it showed why the results must hold from first principles.

Historically, Guldin was accused of plagiarism regarding Pappus’ theorems, since he did not credit Pappus in his work—even though he cited many other sources. In terms of rigor and presentation, Guldin’s proof is “better” than Pappus’, because it provides a logical justification grounded in mechanics rather than intuition. In terms of originality, Pappus deserves credit for formulating the theorems first. Historians often describe Guldin as giving the theorems their first rigorous foundation, while Pappus supplied the original geometric insight (e.g., Mancosu, 1996).

 

In short: Pappus had the idea, Guldin gave it rigor*. This is why the theorems are often referred to as the Pappus–Guldin Theorems in historical and mathematical literature.

 

*An example of Guldin's definition: “A rotation is a simple and perfectly circular motion, around a fixed center, or an unmoved axis, which is called the 'axis of rotation', turning around either a point, or a line, or a plane surface, which, almost as leaving a trace behind it, describes or generates a circular quantity, either a line, or a surface, or a body (Mancosu, 1996, p. 58).”

 

Key Takeway (In Feynman’s style): You see, the trick isn't just that a rotating figure makes a volume. Any fool can see that. The magic is in finding the shortcut. The theorem gives you a way to be lazy, in the very best sense a physicist can be! It’s like this: you don't have to add up every single little piece of the figure. That’s brute force, and it’s messy. Instead, you find the one special point—the centroid—where the whole thing balances. It’s the average location of all the stuff.

Analogy:

· Imagine a team of runners on a circular track. Each runner is at a different lane, tracing a different path around the stadium. Keeping track of all their steps is hopeless.

· Now imagine one “average runner” standing at the centroid position. When this single runner makes a lap, the distance they cover—multiplied by the number of runners—equals the total distance of the whole team.

· That’s Pappus’ trick: instead of handling chaos (calculus) from countless paths, you just follow one “lane” (centroid’s path). One path, multiplied, gives you the total instead of wrestling with the chaos of every path, you just track the path of the centroid. Multiply that single path by the size of the figure, and you’ve got the total.

 

The Moral of the Lesson: Paul Guldin, a Jesuit mathematician, might have smiled at the saying, “It is more blessed to ask forgiveness than permission.” (This so-called Jesuit credo is often used by Nobel laureate Frank Wilczek to justify bold but innovative thinking.) In a well-known controversy, Guldin attacked Cavalieri’s method of indivisibles. Guldin argued that when a surface is generated by the rotation of a line, the surface is not just a collection of lines, and therefore considered Cavalieri’s method flawed. Yet historians favored innovation: Cavalieri’s idea is now recognized as a precursor to integral calculus despite its limitations (Andersen, 1985). Sometimes, what seems boldly ‘wrong’ in the moment may later be perceived as brilliantly right.

 

References:

Andersen, K. (1985). Cavalieri's method of indivisibles. Archive for history of exact sciences, 31(4), 291-367.

Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

Guldin, P. (1641). Centrobaryca seu de centro gravitatis trium specierum quantitatis continuae. Libri IV. Marcus Tudella.

Levi, M. (2009). The mathematical mechanic: using physical reasoning to solve problems. Princeton University Press.

Mancosu, P. (1999). Philosophy of mathematics and mathematical practice in the seventeenth century. Oxford University Press.

Pappus of Alexandria. (1986). Book 7 of the Collection (A. Jones, Ed. & Trans.). Springer-Verlag.