# 95.5.3 Centroid of a composite area

```Centroid of a Composite Area
Steven Vukazich
San Jose State University
Recall the Definition of the
Centroid of an Area
y
y
x
ðķ
ððī
ðĨĖ , ðĶ&quot;
y
x
ðī = * ððī
x
ðĨĖ =
âŽ ðĨððī
First moment of the
ðī
âŽ ðĶððī
ðĶ&quot; =
ðī
First moment of the
If We Can Divide the Area into Simple
Shapes With Known Centroid
y
ðĨ/.
y
ðīð
ðð
ðķ
ðĶ/.
x
ðī = 1 ðī.
ðĨĖ , ðĶ&quot;
x
∑ ðĨ/. ðī.
ðĨĖ =
ðī
∑ ðĶ/. ðī.
ðĶ&quot; =
ðī
First moment of the
First moment of the
Tabulated Centroids of Common Areas
Can be Found in the Textbook
Example Problem
Find the x and y
coordinates of the
area with respect to the
coordinate axes shown.
y
3 in
x
2 in
ðĨĖ , ðĶ&quot;
ðķ
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
ðī3 = 7 3 = 10.5 in2
2
2
ðĨ3 = 7 = 4.67 in
3
1
ðĶ3 = 3 = 1.0 in
3
Find Area and Location of Centroid of Each
Shape Relative to Reference Coordinate Axes
y
ðī@ = 4 4 = 16 in2
Shape 2
ðĨ@ = 5.0 in
5 in
ðĶ@ = −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
4ð
3ð
From the table
c3
2 in
6 in
x
1 @
1
ðīB = − ðð = − ð 2@
2
2
= −6.2832 in2
4 2
ðĨB = 6 −
= 5.1512 in
3ð
ðĶB = 0
Find the x Coordinate of the Centroid
ðĨ3 = 4.67 in
ðĨ@ = 5.0 in
ðĨB = 5.1512 in
ðī3 = 10.5 in2
ðī@ = 16 in2
ðīB = −6.2832 in2
ðī = 1 ðī. = 10.5 + 16 − 6.2832 = 20.2168 in2
∑ ðĨ/. ðī. = 4.67 10.5 + 5.0 16 + 5.1512 −6.2832 = 96.635 in3
∑ ðĨ/. ðī.
96.635 in3
ðĨĖ =
=
= 4.78 in
2
ðī
20.2168 in
Find the y Coordinate of the Centroid
ðĶ3 = 1.0 in
ðĶ@ = −2.0 in
ðī3 = 10.5 in2
ðī@ = 16 in2
ðĶB = 0
ðīB = −6.2832 in2
ðī = 1 ðī. = 10.5 + 16 − 6.2832 = 20.2168 in2
∑ ðĶ/. ðī. = 1.0 10.5 + −2.0 16 + 0 −6.2832 = −21. 5 in3
∑ ðĶ/. ðī.
−21.5 in3
ðĶ&quot; =
=
= −1.06 in
2
ðī
20.2168 in
Coordinates of the Centroid
y
4.78 in
3 in
x
2 in
1.06 in
ðķ
3 in
3 in
4 in
1 in
```