I'm not sure I understand why the radome on the AWACS aircraft is so large, and could somebody explain the design and why it has to rotate?
Basically, the radar antenna of the AWACS system is just like this (source):
Rotating waveguide antenna (picture source)
The round shape is only an aerodynamic fairing. Proof:
Boeing E-3A AWACS cutaway (picture source)
A surveillance antenna can also be fixed. Electronic beam steering helps to view a sector of approximately ±60°, but in order to get a 360° sweep the airplane needs to fly circles. Beam steering is also used on the rotating antenna to rapidly switch between several targets, so this is not unique to a fixed antenna:
Pakistani AWACS based on EMB-145 (picture source)
The alternative would be to have a stationary dome with the radar rotating inside. The Russians did that for their Il-76 AEW aircraft, this is the result:
This dome is noticeably thicker than that of the E-3 because there has to be space for the antenna inside the dome. For the E-3, the outside surface of the antenna is also the outside surface of the dome (specifically, the white band of the dome), meaning the dome can be thinner.
The width of the dome is determined by the width of the antenna. The wider the antenna, the higher the resolution, so you want the largest antenna you can install.
Mounted atop the aircraft fuselage in a rotating dome, the AWACS S-band (E-F band) surveillance radar is able to survey, in 10-second intervals, a volume of airspace covering more than 200,000 square miles (500,000 square km) around the AWACS, or greater than 250 miles (400 km) in all directions
If the radome did not rotate it would only cover the area in the direction the array was pointed (tangential to the stripe). Most similar radomes rotate, although some are enclosed so you may not see the rotating device.
Good answers above.
I just want to add that the planar arrays do not need to fly in circles (or rotate) to provide 360 degree coverage. The beams don't point solely perpendicularly from the radar on either the E-3 (rotating planar array) or E-7A (fixed planar array). They're electronically scanned:
A typical face will cover a 100-120 degree swath of the sky in front of it (a "two-sided" planar radar will cover 200-240 degrees). For fixed/longitudinally installed radars, the main arrays face sideways, and the other 160-120 degrees are often (but not always) covered by smaller radars that face forward and/or aft. These may have less power-range than the primary arrays.
[Boeing E-7 airborne early warning and control aircraft. Credit: Boeing.]
Example: the E-7 above has four main radars, two pointing left/right, two pointing fore/aft:
The 10.8 m long by 3.4 m high antenna assembly incorporates 7.3 m long by 2.7 m high Side-Emitting Electronic Manifold array, with the top hat supporting array providing 120° coverage on port and starboard side, while the top hat array itself provides 60° fore and aft, thus providing a complete 360° coverage. [Wiki. Sorry, I couldn't find an illustration/cutaway.]
So fixed/longitudinal arrays do not necessarily need to fly in constant circles to provide 360 degree coverage. (Note that this is different from saying that the quality/range of coverage is uniform for all bearings. This also isn't saying that AEWC don't fly circuits. The actual tracks can vary and take into account friendly forces and the anticipated threat direction to best situate/point the faces while minding your own security.)
The radome on the AWACS is NOT the black bit. The radar is mounted on the long faces of the metal bit on the diameter, and 'sprays' radar out through the aerodynamic shaping of the black part. Since the radar beam is directional, like most radar beams, it has to be rotated to get 360d coverage.