A vertical array of horizontal full-wave dipoles that are driven by transposed two-conductor line and backed with a parasitic reflecting curtain or horizontal dipoles.
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The size of the dish isn't as important as the length of the antenna. The antenna is INSIDE the little can at the focus of the dish. The length of the antenna is tuned to the wavelength of the frequency being received.
It can be but need not be.
A lot of parabolic reflectors are used in antennas, particularly with frequencies in the microwave region. There are a few other shapes, but they are infrequent choices for antenna designers.
Parallel rays, such as those from a very distance source, are focussed by a parabolic antenna so that they all meet at the focus of the parabola. This results in a stronger signal.
it is the integral of the normalised power pattern over sphere (4pie.st)