Centroid of One-Dimensional and Two-Dimensional Bodies
Concept, Theory, Derivation, Applications and Advanced Cases
The concept of Centroid is one of the most important ideas in Engineering Mechanics, Strength of Materials, Structural Engineering, Machine Design and Fluid Mechanics.
Whenever we deal with beams, plates, laminae, sections, bending stresses, moments of inertia, or stability problems, the centroid becomes fundamental.
In this article, we shall discuss:
One Dimensional (1D) Objects
Two Dimensional (2D) Objects
Concept of Resultant Area
Mathematical Derivation of Centroid
Varignon’s Theorem of Moments
Centroid of Composite Figures
Centroid of Curves and Lines
Symmetry Conditions
Centroid of Common Geometrical Shapes
Negative Areas (Removed Portions)
Centroid by Integration
Applications of Centroid
Relationship Between Centroid and Moment of Inertia
Parallel Axis Theorem
Practical Engineering Examples
1. One Dimensional (1D) Objects
A one-dimensional body possesses only length and negligible thickness.
Examples:
Thin wire
Metallic rod
Straight bar
Circular ring wire
Arc
Practically, a thin metallic rod is treated as a 1D object.
Suppose we have a straight rod of length (L).
If the rod is perfectly uniform, its centroid lies exactly at the midpoint.
For a rod extending from (x=0) to (x=L),
$x_g=\dfrac{L}{2}$
This point represents the balancing point of the rod.
2. Two Dimensional (2D) Objects
A two-dimensional body has length and breadth but negligible thickness.
Such objects are called:
Lamina
Thin plates
Plane areas
Examples:
Rectangular plate
Circular disc
Triangular sheet
Thin metal plates
Suppose we have a thin rectangular plate of dimensions:
Breadth = (b)
Height = (h)
Then its centroid lies at:
$x_g=\dfrac{b}{2},\quad y_g=\dfrac{h}{2}$
3. Physical Meaning of Centroid
The centroid is the geometrical center of an area.
It is the point through which the entire area may be assumed to act.
In simple words:
The centroid is the balancing point of a geometrical figure.
For uniform density materials:
Centroid = Center of Area
Center of Mass = Center of Gravity
all coincide.
4. Concept of Resultant Area
The concept of centroid is developed from the idea of the resultant of several elemental areas.
An area can be divided into infinitely small elemental parts.
Suppose a plane area (A) is divided into small elemental areas:
$a_1, a_2, a_3, \dots a_n$
Each elemental area possesses a position vector in the coordinate plane.
Let:
$(x_1, x_2, x_3,\dots x_n)$ be distances from the Y-axis
$(y_1, y_2, y_3,\dots y_n)$ be distances from the X-axis
Each elemental area produces a moment about the axes.
5. Moment of an Area
The moment of an elemental area about the Y-axis is:
$a_i x_i$
Similarly, moment about X-axis:
$a_i y_i$
The total moment about Y-axis becomes:
$a_1x_1 + a_2x_2 + a_3x_3 + \dots + a_nx_n$
Using sigma notation:
$\sum a_i x_i$
Similarly about X-axis:
$\sum a_i y_i$
6. Varignon’s Theorem of Moments
The entire theory of centroid is based on Varignon’s Theorem.
Statement
The moment of the resultant about any point is equal to the sum of moments of all component vectors about the same point.
Since areas behave like vector systems in moment calculations, the theorem becomes applicable.
Suppose the resultant area (A) passes through a point $G(X_g,Y_g)$.
Then:
Moment of resultant area about Y-axis:
$AX_g$
Moment about X-axis:
$AY_g$
By Varignon’s theorem:
$\sum a_i x_i = AX_g$
and,
$\sum a_i y_i = AY_g$
Since,
$A = \sum a_i$
therefore,
7. Fundamental Centroid Formula
X-coordinate of Centroid
$X_g=\dfrac{\sum a_i x_i}{\sum a_i}$
Y-coordinate of Centroid
$Y_g=\dfrac{\sum a_i y_i}{\sum a_i}$
These are the most fundamental equations of centroid calculations.
8. Continuous Form Using Integration
If the body is continuous, summation changes into integration.
Then:
For Area
$X_g=\dfrac{\int xdA}{\int dA}$
and,
$Y_g=\dfrac{\int ydA}{\int dA}$
9. Centroid of 1D Curves (Lines/Wires)
For wires or arcs:
$dA \rightarrow dL$
Hence,
$X_g=\dfrac{\int xdL}{\int dL}$
$Y_g=\dfrac{\int ydL}{\int dL}$
Used for:
Circular arcs
Wire frames
Curved rods
Bent structures
10. Centroid of Common Geometrical Figures
(a) Rectangle
Dimensions:
Breadth = (b)
Height = (h)
Centroid:
$x_g=\dfrac{b}{2},\quad y_g=\dfrac{h}{2}$
(b) Triangle
For a triangle of height (h):
Centroid lies at:
$y_g=\dfrac{h}{3}$
from the base.
Or,
$\dfrac{2h}{3}$
from the vertex.
(c) Semicircle
For a semicircular area of radius (R):
$y_g=\dfrac{4R}{3\pi}$
from the diameter.
(d) Quarter Circle
$x_g=y_g=\dfrac{4R}{3\pi}$
(e) Circle
By symmetry:
$x_g = 0,\quad y_g = 0$
if origin is at center.
11. Symmetry Conditions
Symmetry greatly simplifies centroid problems.
Case 1: Single Axis Symmetry
If a figure is symmetric about one axis:
centroid lies on that axis.
Case 2: Double Symmetry
If symmetric about both X and Y axes:
centroid lies at their intersection.
Examples:
Circle
Rectangle
Square
12. Composite Bodies
Most engineering structures are not simple shapes.
They are combinations of:
rectangles
circles
triangles
semicircles
holes
Such bodies are called Composite Figures.
13. Algorithm to Find Centroid of Composite Figures
Step 1
Take the complex geometric figure.
Step 2
Divide it into basic standard geometrical shapes.
Step 3
Choose a convenient coordinate system.
All measurements must be based on this frame.
Step 4
For every elemental area determine:
Area $(a_i)$
Centroid coordinates $(x_i,y_i)$
Step 5
Assign signs carefully.
Positive Area
If area is added.
Negative Area
If area is removed (holes/cutouts).
Step 6
Use centroid equations:
$X_g=\dfrac{\sum a_i x_i}{\sum a_i}$
$Y_g=\dfrac{\sum a_i y_i}{\sum a_i}$
14. Negative Area Concept
Suppose:
a circular hole is removed from a plate
a triangular notch is cut
Then removed portions are treated as:
$-a_i$
This is extremely important in engineering design.
15. Example of Composite Area
Suppose a rectangular plate contains:
one circular hole
one triangular cut
Then:
$A = A_1 - A_2 - A_3$
where:
(A_1) → rectangle
(A_2) → circular hole
(A_3) → triangular cut
Then:
$X_g = \dfrac{A_1x_1 - A_2x_2 - A_3x_3}{A_1-A_2-A_3}$
Similarly for (Y_g).
16. Centroid of Three-Dimensional Bodies
For volumes:
$dA \rightarrow dV$
Then centroid becomes center of volume.
Coordinates become:
$X_g=\dfrac{\int xdV}{\int dV}$
$Y_g=\dfrac{\int ydV}{\int dV}$
$Z_g=\dfrac{\int zdV}{\int dV}$
Used in:
solid mechanics
fluid mechanics
ship stability
aerospace structures
17. Applications of Centroid
Centroid is used in:
Beam design
Structural engineering
Machine design
Aircraft design
Stability analysis
Fluid pressure problems
Bending stress analysis
Area moment of inertia
Shear stress distribution
18. Relation Between Centroid and Moment of Inertia
The Moment of Inertia of Area is also called:
Second Moment of Area
It measures resistance against bending.
First Moment of Area
Centroid uses:
$\int xdA$
which is called the first moment of area.
Second Moment of Area
Moment of inertia uses:
$\int x^2dA$
or,
$\int y^2dA$
which is the second moment of area.
19. Centroidal Moment of Inertia
Moment of inertia about centroidal axes is minimum.
Common formulas:
Rectangle
$I_{xx}=\dfrac{bh^3}{12}$
$I_{yy}=\dfrac{hb^3}{12}$
Circle
$I=\dfrac{\pi R^4}{4}$
about diameter.
20. Parallel Axis Theorem
Very important theorem in mechanics.
Statement
Moment of inertia about any axis equals the centroidal moment of inertia plus area times square of distance between axes.
Mathematically:
$I=I_G+Ad^2$
where:
$(I_G)$ = centroidal moment of inertia
(A) = area
(d) = distance between axes
21. Why Centroid is Important Before MOI
Before finding moment of inertia of a complex section:
First find centroid
Locate centroidal axes
Compute centroidal MOI
Transfer using parallel axis theorem
Thus centroid acts as the foundation for advanced structural analysis.
22. Engineering Insight
In engineering mechanics:
Forces produce translational effects
Moments produce rotational effects
Areas produce distributed effects
The centroid represents the effective point through which the entire geometrical effect acts.
That is why centroid becomes one of the most fundamental ideas in mechanics and structural engineering.
23. Final Conclusion
The centroid is the geometrical center of a body obtained from the resultant of all elemental areas. The entire theory is based upon the principles of moments and Varignon’s theorem.
For simple geometries, centroid is obtained directly from symmetry. For complex composite figures, the body is divided into smaller standard figures and the algebraic summation method is used.
The study of centroid forms the basis of:
Area moments
Structural analysis
Beam bending
Stability
Machine design
Fluid mechanics
Advanced mechanics of materials
Thus, centroid is not merely a geometrical concept — it is one of the core foundations of engineering science.