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    Desmos for the SAT · Technique 5 of 15

    Find a Circle's Center and Radius from Its Equation

    Skip completing the square. Graph it and measure.

    SolvesCircle centerCircle radiusCircle equations

    Circle questions love to hand you the expanded equation, like x2 + y2 - 6x + 8y + 9 = 0, and quietly expect you to complete the square twice to dig out the center and radius by hand.

    Here is the shortcut: Desmos graphs that expanded equation exactly as written. You never have to complete the square, you just have to read the picture correctly.

    The key idea: Desmos marks the edges of the circle, but not the center, so you read the edge dots and take the midpoint.

    Nova

    Completing the square twice, just to find a center? Yeah, we're skipping that. Circle questions show up in the Math section, and Desmos graphs the messy equation exactly as written, so you go straight to reading the answer off the picture.

    The algebra way vs the Desmos way

    This technique sidesteps completing the square twice.

    The algebra way

    • Group the x-terms and y-terms and move the constant across: (x^2 - 6x) + (y^2 + 8y) = -9.
    • Complete the square on x by adding 9, and on y by adding 16, to both sides: (x - 3)^2 + (y + 4)^2 = 16.
    • Read the center (3, -4) off the shifts, and the radius as r = sqrt(16) = 4.

    The Desmos way

    • Type the expanded equation x^2 + y^2 - 6x + 8y + 9 = 0 exactly as given.
    • Read the center (3, -4) off the picture and the radius as the distance out to an edge point, 4.

    The payoff: You skip completing the square twice and the sign slips that come with it, and you avoid the classic mistake of reporting r^2 = 16 when the right side of the finished equation is the radius squared, not the radius.

    See the move, live

    That same example, already loaded in the real calculator, the one built into Bluebook on test day. Try the move yourself.

    Desmos graphing calculator, livepre-loaded

    Before you start: set up Desmos

    • Convert decimals to fractions

      Desmos shows decimals like 1.333. SAT choices are usually fractions. Click the round fraction-converter button next to a value to turn 1.333 into 4/3.

    • Turn on Projector Mode for clearer graphs

      Desmos draws thin lines by default, which makes a crossing or intercept hard to pin down. Open the wrench icon (Graph Settings, top right) and turn on Projector Mode for thicker lines and larger labels, so the points you click are easier to hit exactly.

    • Widen the axis bounds when the graph looks empty

      Desmos opens on a small window from -10 to 10 on both axes. If a crossing, vertex, or data point sits outside that box, the graph can look empty even when an answer exists. Open the wrench icon and widen the x and y bounds until the point you need comes into view.

    Step by step

    Nova

    Follow along one step at a time, hit Next as you go. I'll flag the spots where people slip.

    Step 1 of 5

    Type the equation exactly as given, e.g. x2 + y2 - 6x + 8y + 9 = 0.

    Desmos graphing the circle from the expanded equation x^2 + y^2 - 6x + 8y + 9 = 0.

    Syntax: Type the expanded equation directly, e.g. x2 + y2 - 6x + 8y + 9 = 0. Click the circle to reveal its edge points; the radius is the distance from the center to an edge.

    Now try it on a real SAT question

    The equation x2 + y2 + 4x - 12y + 15 = 0 defines a circle in the xy-plane.

    What is the radius of the circle?

    Pick an answer to get instant coaching from Nova.

    Nova

    No center label here. Average left/right for x, top/bottom for y, then measure out to an edge for the radius.

    Your turn, blank calculatornothing loaded

    Set it up yourself in the blank calculator, then read the answer straight off the graph. Muscle memory beats watching.

    Watch it

    Graph the expanded equation and read the center and radius off the picture.

    The Desmos trap: when to close the calculator

    • If the equation is already in center-radius form, (x - h)2 + (y - k)2 = r2, just read off h, k, and r directly. No graphing needed.
    • To check whether a point is inside, outside, or exactly on the circle, plug its coordinates into the equation and compare to r2, rather than eyeballing the graph.
    • Check whether the question wants the radius r, the diameter 2r, or the area pi*r2. Standard form hides r as r2, so read the prompt before you answer.
    • A blank screen usually means a huge radius. Use Zoom Fit to bring the circle back into view.
    • Circles are typed directly as x and y. Do not use a table or regression subscripts.

    Nova

    Gut-check first: if the equation is already in (x - h)2 + (y - k)2 = r2 form, don't even open Desmos. The center and radius are sitting right there, so just read them off and keep moving.

    All Desmos techniques

    Put this move into real practice

    "Find a Circle's Center and Radius from Its Equation" shows up all over the real SAT. SAT Prep Quest drills the skill at your level, so reaching for it becomes automatic by test day.

    Practice free

    SAT is a registered trademark of the College Board, which was not involved in producing, and does not endorse, this product. Score estimates are Learner Labs projections, not official College Board scores.

    Find a Circle's Center & Radius in Desmos | SAT Prep Quest