22 RESISTANCE OF MATERIALS 



16. Centroid of circular sector and segment. To determine the cen- 

 troid of a circular sector OCB (Fig. 14), denote the radius by r and 



the central angle COB by 2 a. 

 Then any small element OPQ 

 of the sector may be regarded 

 as a triangle the centroid of 

 which is on its median at a 

 distance of Jr from 0. The 

 centroids of all these elemen- 

 tary triangles therefore lie on 

 a concentric arc DEFoi radius 

 | r, and the centroid of the 

 entire sector coincides with 

 the centroid of this arc DEF. Therefore, from the results of the pre- 

 ceding article, the centroid of the entire sector OCB is given by 





(16) 



2 sin a 



= r 



3 a 



For a semicircular area of radius r the distance of the centroid 

 from the diameter, or straight side, is 



Sir 



To determine the centroid of a circular segment CBD (Fig. 15), 

 let G denote the centroid of 

 the entire sector OCBD, G Q 

 of the segment CBD, and G 1 

 of the triangle OCD. Then the 

 position of G Q may be deter- 

 mined by noting that the sum 

 of the moments of the triangle 

 OCD and the segment CBD 

 about any point, say 0, is equal 

 to the moment of the entire 

 sector about this point. Thus, 



if A Q , A^ A denote the areas of the segment, triangle, and sector, re- 

 spectively, and # , x^ x, the distances of their centroids from 0, then 



