Some (but not necessarily all) vertices of the shape will lie on the circumcircle. A polygon whose vertices all lie on its circumcircle is said to be a cyclic polygon. All regular simple polygons, all triangles and all rectangles are cyclic.
The circumcircle of a triangle is the unique circle on which all its three vertices lie. The circumcenter of a triangle can be found as the intersection of the three perpendicular bisectors. (A perpendicular bisector is a line that forms a right angle with one of the triangle's sides and intersects that side at its midpoint.) This is because the circumcenter is equidistant from any pair of the triangle's points, and all points on the perpendicular bisectors are equidistant from those points of the triangle.
A triangle is acute (all angles smaller than a right angle) iff the circumcenter lies inside the triangle; it is obtuse (has an angle bigger than a right one) iff the circumcenter lies outside, and it is a right triangle iff the circumcenter lies on one of its sides (namely on the hypotenuse). This is one form of Thales' theorem.
The diameter of the circumcircle can be computed as the length of any side of the triangle, divided by the sine of the opposite angle. (As a consequence of the law of sines, it doesn't matter which side is taken: the result will be the same.) The triangle's nine point circle has half the diameter of the circumcircle.
The circumcenter always lies on one line with the triangle's centroid and orthocenter. This line is known as Euler's line.
The circumcircle is given by the equation
Given
Cyclic polygons
Circumcircles of triangles
Circumcircle equation
where A, B and C are the points of the triangle, and the solution for v is the circumcircle. (Note A²=Ax²+Ay².)
we then have av²-2Sv-b=0, and assuming the three points were not in a line (otherwise the circumcircle doesn't exist), (v-S)²=b/a+S²/a², giving the circumcenter S/a and the circumradius √(b/a+S²/a²). This approach should also work for the circumsphere of a tetrahedron.
See also: incircle.