20 



RESISTANCE OF MATERIALS 



Evidently it is also possible to determine the centroid of an area 

 or line, although neither has a center of gravity or center of mass, 

 since mass and weight are properties of solids. 



For a plane area the centroid is determined by the equations 



^ A VzAa 



(13) A = 2/ Aa > x * = Y ' 



where Aa denotes an element of area and A the total area of the figure. 

 Similarly, for a line or arc the centroid is given by 



(14) 



where A/ denotes an element of length and L the total length of 



the line or arc. 



14. Centroid of triangular area. To find the centroid of a triangle, 



divide it up into narrow strips parallel to one side AC (Fig. 12). 



Since the centroid of each strip PQ is at 

 its middle point, the centroid of the en- 

 tire figure must lie somewhere on the 

 line BD joining these middle points ; that 

 is, on the median of the triangle. Simi- 

 larly, by dividing the triangle up into 

 strips parallel to another side BC, it is 

 proved that the centroid must also lie 

 on the median AE. The point of inter- 

 section G of these two medians must 

 therefore be the centroid of the triangle. 



Since the triangles DEG and ABG are similar, 



DG = DE 

 GB ~ AB ' 



and since DE 1 AB, this gives 



The centroid of a triangle therefore lies on a median to any side at 

 a distance of one third the length of the median from the opposite 

 vertex. From this it also follows that the perpendicular distance 



