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Centroid of a Composite Area StevenVukazich
SanJoseStateUniversity
x
y
Recall the Definition of the Centroid of an Area
𝑦" =∬𝑦𝑑𝐴��𝐴
𝑑𝐴
�̅� =∬𝑥𝑑𝐴��𝐴
𝐴 = *𝑑𝐴�
�
x
y
First moment of the area about the y axis
First moment of the area about the x axis
x
y𝐶�̅�, 𝑦"
𝑦" =∑𝑦./�� 𝐴.𝐴
x
y
𝐴𝑖
�̅� =∑𝑥./�� 𝐴.𝐴
𝐴 =1𝐴.
�
�
x
y
First moment of the area about the y axis
First moment of the area about the x axis
𝐶�̅�, 𝑦"
If We Can Divide the Area into Simple Shapes With Known Centroid
𝑥./
𝑦./𝑐𝑖
Tabulated Centroids of Common Areas Can be Found in the Textbook
x
y
Example Problem
Find the x and ycoordinates of the centroid of the shaded area with respect to the coordinate axes shown.
𝐶�̅�, 𝑦"
3 in
4 in
3 in 3 in 1 in
2 in
Divide Area into Simple Composite Shapes
x
y
3 in
4 in
3 in 3 in 1 in
2 in
+ –1
2
3
Find Area and Location of Centroid of Each Shape Relative to Reference Coordinate Axes
x
y
3 in
7 in
c11 in
4.67 in
Shape1
𝐴3 =127 3 = 10.5in2
𝑥3 =237 = 4.67in
𝑦3 =133 = 1.0in
Find Area and Location of Centroid of Each Shape Relative to Reference Coordinate Axes
x
yShape2 𝐴@ = 4 4 = 16in2
𝑥@ = 5.0in
𝑦@ = −2.0in
4 in
3 in 4 in
c2
5 in
2 in
Find Area and Location of Centroid of Each Shape Relative to Reference Coordinate Axes
x
y Shape3
𝐴B = −12𝜋𝑟@ = −
12𝜋 2@
= −6.2832in2
𝑥B = 6 −4 23𝜋
= 5.1512in
𝑦B = 06 in
2 in
c3
4𝑟3𝜋
From the table
Find the x Coordinate of the Centroid
�̅� =∑ 𝑥./�� 𝐴.𝐴
=96.635in3
20.2168in2= 4.78in
𝐴 =1𝐴. = 10.5 + 16 − 6.2832 = 20.2168in2�
�
𝐴3 = 10.5in2
𝑥3 = 4.67in
𝐴@ = 16in2
𝑥@ = 5.0in
𝐴B = −6.2832in2
𝑥B = 5.1512in
∑ 𝑥./�� 𝐴. = 4.67 10.5 + 5.0 16 + 5.1512 −6.2832 = 96.635in3
Find the y Coordinate of the Centroid
𝑦" =∑𝑦./�� 𝐴.𝐴
=−21.5in3
20.2168in2= −1.06in
𝐴 =1𝐴. = 10.5 + 16 − 6.2832 = 20.2168in2�
�
𝐴3 = 10.5in2 𝐴@ = 16in2 𝐴B = −6.2832in2
∑𝑦./�� 𝐴. = 1.0 10.5 + −2.0 16 + 0 −6.2832 = −21. 5in3
𝑦3 = 1.0in 𝑦@ = −2.0in 𝑦B = 0
Coordinates of the Centroid
x
y
𝐶
3 in
4 in
3 in 3 in 1 in
2 in
4.78 in
1.06 in
