Centroid (centre of gravity) of Cardboard Quadrilateral

Investigate
Where is the centroid (balancing point or centre of gravity) of a cardboard quadrilateral (a two-dimensional shape (lamina) of uniform density) located?

Centroid (centre of gravity) of Cardboard Quadrilateral

Instructions
Method 1
a) To find the centroid (centre of gravity or balancing point) of a cardboard quadrilateral, divide the quadrilateral by drawing a diagonal and connect the centroids of the two formed triangles with a line - Use the tools on the left (scroll down, if necessary) to do these constructions or click on the Step 1 button.
b) Next draw the other diagonal and repeat the process. Clearly, the centroid (balancing point) of the cardboard quadrilateral now has to lie somewhere on the one line, as well as on the other. Hence, it has to be at their intersection - Use the tools on the left (scroll down, if necessary) to do these constructions or click on the Step 2 button.
Method 2
c) Click on the 'Link to Second Method' button to navigate to a new sketch. Since the relative weights of the cardboard triangles ABD and BCD are determined by their areas, we can determine the relative weights, respectively, concentrated at the centroids G3 and G4 by measuring the areas of triangles ABD and BCD.
d) Now using the lever law of Archimedes, we can easily determine the balancing point (centroid) of the weights (areas concentrated) at G3 and G4. In particular, dilating G3 from G4 as centre and a scale factor of areaABD/(areaBCD + areaABD) gives us the required centroid (also see De Villiers & Jahnke, 2024).
e) Use the tools on the left (scroll down, if necessary) to do this calculation and dilation. Click on the 'Show Cardboard centroid' button to check your construction.
Method 3
Miller (2009) describes an interesting different construction method than the two methods given above for finding the cardboard (lamina) centroid of a quadrilateral.
Note: In physics terms, a 'cardboard' figure is assumed here as being a two-dimensional shape (lamina) of uniform density.

Experimental Testing
Cut out a few irregular shaped quadrilaterals from cardboard, then find their balancing points as described above. Next check with a flat tipped pen or sharp pointed eraser that they indeed balance at the constructed points.

Note: The centroid of a cardboard quadrilateral (a planar quadrilateral of uniform density), unlike the case for a triangle, does NOT always coincide with the point mass (vertex) centroid when equal masses or weights are placed at the vertices - click on the Show Point Mass Centroid button. Also see the dynamic geometry sketch at Point Mass Centroid of Quadrilateral for more information.

Parallelogram Centroid Theorem
Regarding the above Note, the following interesting & important theorem holds:
The point mass (vertex) centroid G and the cardboard centroid of a quadrilateral coincides, if and only if, the quadrilateral is a parallelogram.
Challenge: First try to prove it yourself - before reading the translation from German of a neat proof by Arnold Kirsch in Humenberger (2023). An alternative solution is also provided in Kim et al (2016).

Bisect-diagonal Quadrilateral Theorem
A 'bisect-diagonal' quadrilateral is a quadrilateral with at least one of its diagonals bisecting the other. The following interesting theorem in relation to a bisect-diagonal quadrilateral holds:
If ABCD is a bisect-diagonal quadrilateral with diagonal BD bisected (cut in half) by diagonal AC, then the cardboard (lamina) centroid and point mass centroid both lie on diagonal AC. More-over, if P is the midpoint of AC, then the distance between the cardboard centroid and the point P is twice that of the distance between the cardboard centroid and point mass centroid (for a dynamic sketch see Bisect-diagonal Quadrilateral Theorem).
Challenge: First try to prove it yourself - before reading De Villiers (2021).

Explore Further
1) Is the centroid of a cardboard (lamina) quadrilateral always inside? Specifically check by dragging until the quadrilateral becomes concave.
2) Can you figure out a way of finding the lamina centroid of a cardboard pentagon, hexagon, etc. by dividing them up into suitable triangles, quadrilaterals, etc.?
3) Where is the point mass or vertex centroid of a quadrilateral located if equal weights are placed at the vertices?
4) Where would you locate the balancing point of a 'perimeter' quadrilateral? For example, of a quadrilateral consisting of just sticks, rods or rigid wire forming its perimeter?
5) For what type of cyclic quadrilateral would its lamina centroid coincide with the circumcentre?
(Hint: Have a look at one of the External Links below).

References & Readings
De Villiers, M. (2021). Some more properties of the bisect-diagonal quadrilateral. The Mathematical Gazette, Volume 105, Issue 564, November, pp. 474 - 480.
De Villiers, M. & Jahnke, H.N. (2024). Using Scientific Principles in Mathematics, pp. 2570-2576, Excerpt from: "The Intimate Interplay Between Experimentation and Deduction: Some Classroom Implications", pp. 2551-2586. In Sriraman, B. (Ed.). Handbook of the History and Philosophy of Mathematical Practice, Springer Nature, Switzerland. DOI: https://doi.org/10.1007/978-3-031-40846-5.
Humenberger, Hans. (2023). Centroids of Quadrilaterals and a Peculiarity of Parallelograms. At Right Angles, November, pp. 1-9.
Kim, D-S.; Lee, K.S.; Lee, K.B.; Lee, Y.I.; Son, S; Yang, J.K. & Yoon, D.W. (2016). Centroids and Some Characterizations of Parallelograms. Commun. Korean Math. Soc. 31, No. 3, pp. 637–645. DOI: http://dx.doi.org/10.4134/CKMS.c150165
Miller, J.B. (2010). The centroid of a quadrilateral. Gazette, Australian Mathematical Society.
Shilgalis, T.W. & Benson, C.T. (2001). Centroid of a Polygon—Three Views. The Mathematics Teacher, Vol. 94, No. 4 (April), pp. 302-307.
v. Baravalle, H. (1947). Centroids. The Mathematics Teacher, 40(5), 241–249. From: http://www.jstor.org/stable/27953225.

External Links
Centroid (Wikipedia)
A Simple Lever (Wolfram Demonstration Project)
Dilation (geometry) (Wikipedia)
When is the centroid of a cyclic quadrilateral also its circumcenter? (Ocean of Math)
Nine-point circle (Wikipedia)
Euler line (Wikipedia)
Spieker circle (Wikipedia)
Nagel Line (Wolfram MathWorld)
Napoleon's theorem (Wikipedia)

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Created by Michael de Villiers, with JavaSketchpad, 6 April 2010; updated to WebSketchpad, 16 October 2021; 16 March 2024; 23 Nov 2025.