Clipping the circles can be done using the editing tools. First, select the innermost circle and click the 'clip' command under the 'Editor' menu. Then select the middle circle (which will now actually be an annulus) and click the 'clip' command again.
For the sectors, you should first make a new shapefile of 8 isosceles triangles radiating out from the center point. You can then intersect the triangles with the annuli to arrive at your final shapes.
The coordinates for the points of your first triangle (clockwise from north) will be:
x_1 = x_circle_center (x-coordinate of the center of the circles)
y_1 = y_circle_center
x_2 = (x_circle_center + cos(0 * pi / 180) * length)
y_2 = (y_circle_center + sin(0 * pi / 180) * length)
x_3 = (x_circle_center + cos(45 * pi / 180) * length)
y_3 = (y_circle_center + sin(45 * pi / 180) * length)
For the second triangle you will use 45 and 90 in place of 0 and 45, and so on.
The length that you use must go sufficiently past the outside of the outer annulus so that the shortest leg of the triangle does not cut into it. Just use something like 1.5 times the longest radius (in your example, 1.5 * 80 km = 120 km), which will be way more than enough.
All of this can be implemented in Python using arcpy geometries, but you can also do this manually using the editing tools in ArcMap. Just calculate the coordinates for your triangles and enter them into the Edit Sketch Properties window.
And be sure to use a projected coordinate system for this (not latitude and longitude). If you want to be very accurate with your angles and distances, you could reconstruct your circles and create your triangles in an azimuthal equidistant projection centered on the circle center. Depending on your needs, that might not be necessary.