The area of standard stop sign regular octagon is 742.3 squared inches, if the standard stop sign has a 16.2 inch radius.
Step-by-step explanation:
The given is,
                Radius of standard stop sign is 16.2 inch radius
Step:1
               The regular octagon is equal to the 16 right angled triangle,
                     Angle of right angle triangle  =  = 22.5°
 = 22.5°
               Ref attachment,
               From the OAB right angle triangle,  
               Trigonometric ratio,
                                  sin ∅ =  
        
               Where, ∅ = 22.5°
                            Hyp = Radius = 16.2 inches
               Ratio becomes,
                            sin 22.5°= 
                                        b = (0.374607)(16.2)
                                        b = 6.19947 inches
              Trigonometric ratio,
                                  cos ∅ =  
        
               Where, ∅ = 22.5°
                            Hyp = Radius = 16.2 inches
                             Adj = h
               Ratio becomes,
                            cos 22.5°= 
                                        h = (0.9238795)(16.2)
                                        h = 14.967 inches
Step:2
            From the triangle OAC,
                              Area, 
           Where, Height = 14.967 inches
                        Base = b + x = 6.19947 + 6.19947
                                              = 12.3989 inches
            From the values equation becomes,
                                        A = 
                                        A = 92.7875 Squared inches
Step:3
            The octagon is equal to sum of 8 triangles
             Area of octagon = 8 × Area of triangle
                                         = 8 × 92.7875
                                         = 742.3 squared inches
Result:
         The area of standard stop sign regular octagon is 742.3 squared inches, if the standard stop sign has a 16.2 inch radius.