how to plot a circle (without circle())

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There are several ways you can use plot or similar functions.

• use plot with the circle as the graphics object you already discovered:

  sage: plot(circle((0, 0), 2))

• express y as a function of x for two half-circles (this will miss part of the circle):

  sage: p = plot(sqrt(4 - x^2), (-3, 3))
  sage: q = plot(-sqrt(4 - x^2), (-3, 3))
  sage: (p + q).show(aspect_ratio=1)

• use an implicit plot starting from an equation for the circle:

  sage: x, y = SR.var("x y")
  sage: implicit_plot(x^2 + y^2 - 4, (x, -3, 3), (y, -3, 3))

• use a parametric plot for the circle; here is one (there are many others):

  sage: t = var('t')
  sage: parametric_plot((2*cos(t), 2*sin(t)), (t, 0, 2*pi))

In this case I would recommend the parametric plot rather than the functions or the implicit plot.