Define a frame of reference and a point of origin. For example if asked to find the centroid of a rectangle draw the x and y coordinate system, the rectangle and point (0,0).
Label the coordinates of the four points that define the rectangle. For example (1,1) (5,1) (1,3) (5,3).
Find the midpoints of the vertical segments of the defined rectangle. The coordinates of the midpoint are given by: x = (x1 + x2) /2 and y = (y1 + y2) /2. In the example the segments in question are defined by (1,1) (1,3) and (5,1) (5,3). Therefore the two midpoints are given by, x = (1 + 1)/2 or x = 1 and y = (1 + 3)/2 or y = 2. T he first midpoint needed is at (1,2) and the second is given by x = (5 + 5)/2 or x = 5 and y = (1 + 3)/2 or y = 2. The second midpoint needed is at the point (5,2).
Draw a line segment connecting the two midpoints of the vertical segments.
Find the midpoints of the horizontal sides of the defined rectangle. For example, if the horizontal sides of a rectangle were defined by the points (1,1) (5,1) and (1,3) (5,3), the midpoints would be as follows: x = (x1 + x2)/2, y = (y1 + y2)/2 or x = (1 + 5)/2, y = (1 + 1)/2 so the point (3,1) is the first midpoint. For the second horizontal line, x = (1 + 5)/2 or x = 3, y = (3 + 3)/2 or y = 3. The second midpoint is (3,3).
Draw a line segment connecting the midpoints of the horizontal segments.
Mark the centroid of the rectangle. It will be where the line segments originating at the midpoints of the vertical and horizontal sides of the rectangle intersect.
Pick a frame of reference and a point of origin, for example the x-y coordinate plane and the origin (0,0).
Break the complex object into smaller more manageable objects.
Find the total area by finding the areas of the sub-regions and adding them. For example if given an object that when broken into sub-regions yielded area 1, a rectangle, bounded by points (20,0)(60,0)(60,60) and (20,60) A1 = length (l) x width (w) or 40 mm x 60 mm = 2400 mm^2. A second sub-region bounded by (0,60)(0,70)(80,60) and (80,70) yields A2 = 80 mm x 10 mm = 800 mm^2. Total Area (A(total)) = A1 + A2 = 2400 mm^2 + 800 mm^2 = 3200 mm^2.
Calculate the first moment of areas Q(x1) and Q(x2) relative to the x-axis and add the results to find the first moment of the entire area, with respect to the x-axis, Q(xtotal).
Q(xtotal) = Q(x1) + Q(x2) where:
Q(x1) = the moment of area 1 relative to the x-axis
Q(x2) = the moment of area 2 relative to the x-axis
Q(x1) = y1A1 where:
y1 = distance from y-axis to center of area 1
A1 = calculated area of area 1
Q(x2) = y2A2 where:
y2 = distance from y-axis to center of area 2
A2 = calculated area of area 2
Q(x1) = 30 mm x 2400 mm^2
Q(x1) = 72000 mm^3
Q(x2) = 65 mm x 800 mm^2
Q(x2) = 52000 mm^3
Q(xtotal) = 72000 mm^3 + 52000 mm^3
Q(xtotal) = 124000 mm^3
Calculate the first moment of areas Q(y1) and Q(y2) relative to the y-axis and add the results to find the first moment of the entire area, with respect to the y-axis, Q(ytotal).
Q(ytotal) = Q(y1) + Q(y2) where:
Q(y1) = the moment of area 1 relative to the y-axis
Q(y2) = the moment of area 2 relative to the y-axis
Q(y1) = x1A1 where:
x1 = distance from x-axis to center of area 1
A1 = calculated area of area 1
Q(y2) = x2A2 where:
x2 = distance from x-axis to center of area 2
A2 = calculated area of area 2
Q(y1) = 40 mm x 2400 mm^2
Q(y1) = 96000 mm^3
Q(y2) = 40 mm x 800 mm^2
Q(y2) = 32000 mm^3
Q(ytotal) = 96000 mm^3 + 32000 mm^3
Q(ytotal) = 128000 mm^3
Find the X and Y coordinates of the centroid of the entire region.
Y =Q(xtotal) / A(total) and X = Q(ytotal) / A(total) where:
Y = y coordinate of centroid
X = x coordinate of centroid
Q(xtotal) = sum of the first moment of the areas relative to the x-axis
A(total) = sum of areas of all sub-regions
Q(ytotal) = sum of the first moment of the areas relative to the y-axis
Y = 124000 mm^3 / 32000 mm^2
Y = 3.875 mm
X = 128000 mm^3 / 32000 mm^2
X = 4 mm
X and Y coordinates of centroid = (4, 3.875)