Math Learning

Circle Equations and Conic Section Explorer

Start with circle equations, then compare how the squared terms create an ellipse, parabola, or hyperbola.

Explore conic section graphs, circle equations, hyperbola, parabola, and ellipse in one interactive tool.

Big idea: A circle is one type of conic. When both squared terms act equally, you get a circle. When they stretch, disappear, or have opposite signs, you see an ellipse, parabola, or hyperbola.

Circle form: both squared terms are positive and balanced.

ConicCircle
Center / vertex(0, 0)
Key measurer = 4
General form for selected conic(x - h)2 + (y - k)2 = r2Equation with current values(x - 0)2 + (y - 0)2 = 16This is a circle because x2 and y2 have equal positive weight.
What changed in the equation?

    How the equation chooses the shape

    Circle

    x2 and y2 are both positive and balanced.

    Ellipse

    x2 and y2 are both positive, but stretched differently.

    Parabola

    Only one variable is squared in the standard form.

    Hyperbola

    The squared terms have opposite signs, so the graph opens in two branches.

    Practice keywords

    Use the presets first, then move one slider at a time and describe how the equation changes the graph.

    conic sectioncircle equationsellipseparabolahyperbolaconic sections