Is also called a segment.
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You divide the area of the shaded region by the area of the full circle. For example, if the radius of the shaded region is 2 meters, the probability would be 4pi / 36pi, or 1/9. If the shaded region is a 'slice' of the circle, the chance is just the fraction of the circle which the 'slice' is.
A cross section, I believe
It is a sector of the circle
They are the shapes of the slices when you slice a cone. For example, when you slice it parallel to the base and look at the shape of the slice, you see the conic section known as a "circle". The others are the "ellipse", the "parabola", and the "hyperbola". Which one you get depends only on how you tilt the knife when you slice the cone.
It is a sector