QGIS calculate radius of an arc to attributes

Lightly tested formulas follow, so proceed with caution. But following along with an example here: https://www.mathopenref.com/arcradius.html

If your circular arcs have a vertex at the middle point along the arc (which I am saying is x1, y1 in the figure), you could use it along with the start and end points to calculate the chord length "W" and the height "h" to get the radius "R" using the following formula:

and saying

W =

sqrt(
     ( $x_at(-1) - $x_at(0) )^2 +
     ( $y_at(-1) - $y_at(0) )^2
    )

and

H =

sqrt(
    ( $x_at(1) - ( $x_at(-1) + $x_at(0) )/2 )^2
    + ( $y_at(1) - ( $y_at(-1) + $y_at(0) )/2 )^2
)

in your Expression Dialog of the Field Calculator you'd have this long equation for the radius calculation:

R =

sqrt( 
    ( $x_at(1) - ($x_at(-1) + $x_at(0) )/2 )^2 
    + ( $y_at(1) - ($y_at(-1) + $y_at(0))/2 )^2
     ) /2
+
(   ( $x_at(-1) - $x_at(0) )^2 )  + 
    ( $y_at(-1) - $y_at(0) )^2 )  )
/ ( 8 * sqrt(
            ( ($x_at(1) - ($x_at(-1) + $x_at(0) )/2 )^2
            + ( $y_at(1) - ($y_at(-1) + $y_at(0) )/2 )^2
             )
   )