Consider one of 5 isosceles triangles that make this pentagon.


The base of this triangle is 2 inches, hence, half of the base is 1 inch long.


The angle of the triangle, opposite to the base, is  360/5 = 72 degrees;
half of this angle is 36 degrees.


Let r be the radius of the circle, inscribed in this pentagon (in inches).


Then   = tan(36).


Hence,  r =  =  = 1.3763819 inches.


For tan(36), there is special formula  tan(36) = .


For deriving this formula, see, for example, this source

    https://www.cuemath.com/trigonometry/tan-36-degrees/

or many others (it is a classic in Trigonometry).



So, the answer to your question for the diameter is  

    D =  =  =  2.752763841 inches.