Radius of Gyration Definition
Formula  Transfer of Axis 


The Radius of Gyration of an Area about a given axis is a distance k from the axis. At this distance k an equivalent area is thought of as a line Area parallel to the original axis. The moment of inertia of this Line Area about the original axis is unchanged.  


The Radius of Gyration of a Mass about a given axis is a distance k from the axis. At this distance k an equivalent mass is thought of as a Point Mass. The moment of inertia of this Point Mass about the original axis is unchanged.  




Figure 1  
It is possible for two EQUAL areas to have equal MOMENT'S OF INERTIA about the same axis while having different dimensions.  
Figure 1 contains two rectangles. Setting the Area's and Moment's of Inertia equal, the following relationships can be found.  
equ. (1)  
equ. (2)  
h_{1} and b_{1} are constants.  
b_{2} depends on h_{2}  
d(h_{2}) is: 

equ. (3)  
In equ. (3) there is a limited range of h_{2} that yields a real solution. d is maximized when h_{2} goes to zero. This corresponds to an infinitely long, infinitely thin rectangle. The maximum value of d is the Radius of Gyration. 

equ. (4)  
This is equivalent to using the general Radius of Gyration equation on a rectangle.  
The mass Radius of Gyration is the same concept just applied to a mass. 