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Digital SAT · Solve it in Desmos

How to Solve Circle Equations on Desmos (Digital SAT)

Graphing is fast and error-proof for general-form equations where completing the square is fiddly, and it removes the sign-flip mistake. Read from standard form by hand when the equation already sits in that form.

Perfect1600 Content Team·Uses the same Desmos calculator that is built into the Digital SAT

Full method for circle equations

The Desmos steps

  1. 1

    Type the equation

    Enter it in standard or general form and Desmos draws the circle.

  2. 2

    Read center and radius

    See the center and click a point on the circle; the distance to the edge is the radius.

  3. 3

    Answer the asked value

    Take the center, radius, or diameter from the graph.

Open a real question in the same Desmos calculator you get on Perfect1600, with the setup already typed in:

Worked in Desmos

Easy example
The graph of (x−6)2+(y−3)2=81(x - 6)^{2} + (y - 3)^{2} = 81 is a circle. What is the radius of this circle?
A
3
B
6
9
D
81

A: Incorrect. This is one coordinate of the center, not the radius.

B: Incorrect. This is the other coordinate of the center, not the radius.

C: Correct. r2=81r^{2} = 81, so r=81=9r = \sqrt{81} = 9.

D: Incorrect. This is r2r^{2}, not the radius itself.

Explanation

Since r2=81r^{2} = 81, the radius is 81=9\sqrt{81} = 9.

Easy example
The graph of x2+y2=36x^{2} + y^{2} = 36 is a circle. What is the radius of this circle?
6
B
9
C
12
D
18

A: Correct. r2=36r^{2} = 36, so r=36=6r = \sqrt{36} = 6.

B: Incorrect. This does not equal 36\sqrt{36}.

C: Incorrect. This is twice the radius, the diameter.

D: Incorrect. This triples the radius.

Explanation

Since r2=36r^{2} = 36, the radius is 36=6\sqrt{36} = 6.

Easy example
A circle in the xyxy-plane has equation (x+8)2+(y+2)2=144(x+8)^2+(y+2)^2=144. Which of the following points lies on this circle?
A
(3,−2)(3,-2)
B
(4,−3)(4,-3)
(4,−2)(4,-2)
D
(4,0)(4,0)

A: Incorrect. This uses an xx-coordinate one less than correct: (3,−2)(3,-2) does not satisfy the equation.

B: Incorrect. This uses a yy-coordinate one less than correct: (4,−3)(4,-3) does not satisfy the equation.

C: Correct. Starting at the center (−8,−2)(-8,-2) and moving 12 units right gives (−8+12,−2)=(4,−2)(-8+12,-2)=(4,-2), which satisfies (4+8)2+(−2+2)2=144+0=144(4+8)^2+(-2+2)^2=144+0=144.

D: Incorrect. This uses y=0y=0 instead of the center's yy-coordinate: (4,0)(4,0) does not satisfy the equation.

Explanation

Starting at the center (-8, -2) and moving 12 units (the radius) right gives (4, -2), which satisfies the equation.

Try it with the calculator

Question 1easy

The graph of (x−4)2+(y+1)2=25(x - 4)^{2} + (y + 1)^{2} = 25 is a circle. What is the center of this circle?

Question 2easy

The graph of (x−2)2+(y−5)2=49(x - 2)^{2} + (y - 5)^{2} = 49 is a circle. What is the diameter of this circle?

Question 3 is ready when you are

Keep going with more questions of this type, each with the same step-by-step reasoning. Free to start.

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Desmos questions

Can Desmos solve circle equations questions on the SAT?

Graphing is fast and error-proof for general-form equations where completing the square is fiddly, and it removes the sign-flip mistake. Read from standard form by hand when the equation already sits in that form.

Is the Desmos calculator really built into the Digital SAT?

Yes. The Bluebook testing app includes the Desmos graphing calculator on every Math question, so the method here is one you can use on test day, not a workaround.

Should I still learn the algebra?

Yes. Desmos is fastest when you know what to type and what to read. Learn the underlying method on the full guide, then use the calculator to move quickly and to check your work.

Master circle equations, calculator and all

Every Digital SAT question type comes with the method, the Desmos shortcut, and free practice with instant feedback.

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