Desmos for the SAT · Technique 6 of 23

Find a Circle's Center and Radius from Its Equation

Skip completing the square. Two dots and two functions give the center and the radius, with no arithmetic of your own.

SolvesCircle centerCircle radiusCircle equations

Watch the lesson

The whole move worked on one real SAT question, narrated.

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A second circle, x^2 + y^2 - 10x + 6y - 15 = 0, worked the same way: both dots clicked, then distance over 2 for the radius and mean for the center.

Circle questions usually give you the expanded equation, like $x^{2} + x + y^{2} + y = 199/2$ , and expect you to complete the square twice to find the center and radius. This one is a real Hard question from the College Board, and by hand it means halving a 1, squaring the half, doing it twice, and carrying a fraction on the right.

Desmos graphs that expanded equation exactly as written. Select it and gray dots appear at the very top and the very bottom of the circle, directly above and below the center. Hand both points to midpoint( and Desmos prints the center; distance( over 2 gives the radius. No completing the square, and no arithmetic of your own. Desmos does not mark the leftmost and rightmost points, so work vertically.

Nova

Completing the square twice, just to find a center? Skip that. Desmos graphs the messy equation exactly as written and marks the top and bottom of the circle for you. Click two dots, average two numbers, done.

Now you try it

The rest of the page takes that move slowly. First how it compares with doing the algebra, then the steps in order, then a real SAT question with a live Desmos calculator to try it in.

The algebra way vs the Desmos way

This technique sidesteps completing the square twice.

The question

$x^{2} + x + y^{2} + y = 199/2$

What is the length of the circle's radius?

The algebra way

Complete the square on x: half of 1 is 1/2, so x^2 + x = (x + 1/2)^2 - 1/4.
Do it again on y, then move both -1/4 terms across: 199/2 + 1/4 + 1/4 = 100.
Only now can you read r = sqrt(100) = 10, after fractions in three separate places.

The Desmos way

Type x^2 + x + y^2 + y = 199/2 exactly as given, then click it to bring up the dots.
Click the top and bottom dots: (-0.5, 9.5) and (-0.5, -10.5).
Type distance(top, bottom)/2 and read 10. No fractions touched.

The payoff: Two clicks replace completing the square twice over fractions, which is where this question is designed to lose you.

Step by step

The whole move, all 5 steps

Type the circle equation exactly as given. Desmos draws it with no completing the square.
Click inside the equation box, or on the circle itself, to reveal the gray dots at the top and the bottom.
Click the top dot, then click the export button beside its coordinates so the point stays on screen as its own line.
Click back inside the equation box to bring the dots back, then do the same for the bottom dot.
Type midpoint(top, bottom) with both points and Desmos prints the center; distance(top, bottom) over 2 gives the radius. No arithmetic of your own. mean(top, bottom) returns the same point, and the video uses that name.

The question

The graph of $x^{2} + x + y^{2} + y = 199/2$ in the xy-plane is a circle.

What is the length of the circle's radius?

A9.5
B10
C20
D100

Step 1 of 5

Type the equation exactly as given, e.g. $x^{2} + x + y^{2} + y = 199/2$ . Desmos draws the circle without any rearranging.

In the calculator: x^2+x+y^2+y=199/2

Straight in, no setup. Desmos takes the expanded form happily, which is already more than a handheld calculator will do for you.

Syntax: Type the expanded equation directly, e.g. $x^{2} + x + y^{2} + y = 199/2$ . The gray dots only appear once the expression is selected, so click the equation box (or the circle) first, then click a dot to read its coordinates.

Now try it on a real SAT question

Head or scratch paper. Standard form hands you the radius: 25 is r squared, so $r = 5$ . Nothing to graph, nothing to complete. When the equation already looks like this, close the calculator.

A circle in the xy-plane has equation $(x - 3)^{2} + (y + 2)^{2} = 25$ .

What is the diameter of the circle?

Pick an answer to get instant coaching from Nova.

Stuck?

Your turnstarts empty

Set this question up yourself, then read the answer straight off the graph.

Worked it out on scratch paper? Type it into Desmos anyway and check. It takes a couple of seconds and it catches the slip you cannot spot by re-reading your own work.

The Desmos trap: when to close the calculator

If the equation is already in center-radius form, (x - h)² + (y - k)² = $r^{2}$ , read off h, k, and r directly. No graphing needed.
Check whether the question wants the radius r, the diameter 2r, or the area pi* $r^{2}$ . Standard form hides r as $r^{2}$ , so read the prompt before you answer.
To check whether a point is inside, outside, or exactly on the circle, plug its coordinates into the equation and compare to $r^{2}$ , rather than eyeballing the graph.
Circles are typed directly as x and y. Do not use a table or regression subscripts.

If the calculator does something you did not expect

A blank screen usually means a big radius. Open the wrench icon and widen the x and y bounds until the circle comes into view, since Zoom Fit only appears for tables, lists and points, not for an implicit equation like this one.
If the circle comes out looking like an oval, the axes are on different scales. Open the wrench icon and set the x and y bounds to spans of the same size, e.g. both 12 wide. It matters here, because a squashed circle makes the top and bottom dots harder to trust by eye.
Desmos marks the top and bottom of a circle, and where it crosses an axis, but never the center and never the leftmost or rightmost points. Read it vertically: the sides will not help you.

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.

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