Centroid of a Composite Area
Steven Vukazich
San Jose State University
Recall the Definition of the
Centroid of an Area
y
y
x
𝐶
𝑑𝐴
𝑥̅ , 𝑦"
y
x
𝐴 = * 𝑑𝐴
x
𝑥̅ =
∬ 𝑥𝑑𝐴
First moment of the
area about the y axis
𝐴
∬ 𝑦𝑑𝐴
𝑦" =
𝐴
First moment of the
area about the x axis
If We Can Divide the Area into Simple
Shapes With Known Centroid
y
𝑥/.
y
𝐴𝑖
𝑐𝑖
𝐶
𝑦/.
x
𝐴 = 1 𝐴.
𝑥̅ , 𝑦"
x
∑ 𝑥/. 𝐴.
𝑥̅ =
𝐴
∑ 𝑦/. 𝐴.
𝑦" =
𝐴
First moment of the
area about the y axis
First moment of the
area about the x axis
Tabulated Centroids of Common Areas
Can be Found in the Textbook
Example Problem
Find the x and y
coordinates of the
centroid of the shaded
area with respect to the
coordinate axes shown.
y
3 in
x
2 in
𝑥̅ , 𝑦"
𝐶
3 in
3 in
4 in
1 in
Divide Area into Simple Composite Shapes
2
y
–
+
1
3
3 in
x
2 in
4 in
3 in
3 in
1 in
Find Area and Location of Centroid of Each
Shape Relative to Reference Coordinate Axes
y
Shape 1
c1
4.67 in
7 in
1 in
3 in
x
1
𝐴3 = 7 3 = 10.5 in2
2
2
𝑥3 = 7 = 4.67 in
3
1
𝑦3 = 3 = 1.0 in
3
Find Area and Location of Centroid of Each
Shape Relative to Reference Coordinate Axes
y
𝐴@ = 4 4 = 16 in2
Shape 2
𝑥@ = 5.0 in
5 in
𝑦@ = −2.0 in
c2
3 in
4 in
2 in
x
4 in
Find Area and Location of Centroid of Each
Shape Relative to Reference Coordinate Axes
y
Shape 3
4𝑟
3𝜋
From the table
c3
2 in
6 in
x
1 @
1
𝐴B = − 𝜋𝑟 = − 𝜋 2@
2
2
= −6.2832 in2
4 2
𝑥B = 6 −
= 5.1512 in
3𝜋
𝑦B = 0
Find the x Coordinate of the Centroid
𝑥3 = 4.67 in
𝑥@ = 5.0 in
𝑥B = 5.1512 in
𝐴3 = 10.5 in2
𝐴@ = 16 in2
𝐴B = −6.2832 in2
𝐴 = 1 𝐴. = 10.5 + 16 − 6.2832 = 20.2168 in2
∑ 𝑥/. 𝐴. = 4.67 10.5 + 5.0 16 + 5.1512 −6.2832 = 96.635 in3
∑ 𝑥/. 𝐴.
96.635 in3
𝑥̅ =
=
= 4.78 in
2
𝐴
20.2168 in
Find the y Coordinate of the Centroid
𝑦3 = 1.0 in
𝑦@ = −2.0 in
𝐴3 = 10.5 in2
𝐴@ = 16 in2
𝑦B = 0
𝐴B = −6.2832 in2
𝐴 = 1 𝐴. = 10.5 + 16 − 6.2832 = 20.2168 in2
∑ 𝑦/. 𝐴. = 1.0 10.5 + −2.0 16 + 0 −6.2832 = −21. 5 in3
∑ 𝑦/. 𝐴.
−21.5 in3
𝑦" =
=
= −1.06 in
2
𝐴
20.2168 in
Coordinates of the Centroid
y
4.78 in
3 in
x
2 in
1.06 in
𝐶
3 in
3 in
4 in
1 in