Digital SAT · Circles

How to Solve Circle Equations on the Digital SAT

In standard form, (x - h) squared plus (y - k) squared equals r squared, a circle wears its center and radius openly: the center is (h, k) and the radius is the square root of the right side. Two things to respect: the signs flip, so (x - 6) puts the center's x at positive 6, and the right side is r squared, so an 81 means a radius of 9. If the equation is expanded, complete the square, or simply graph it in Desmos.

Written from Perfect1600’s analysis of every circle equations question in our bank·Method checked against the current Bluebook test

per test: 1 per test
typical difficulty: Medium to hard
practice questions: 74

How we counted

Frequency reflects how often this question type appears on a full-length Digital SAT. The difficulty mix reflects every question of this type across our bank.

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What the question bank shows

read it

Signs flip, right side is r²

The center takes the opposite sign; the radius is the square root of the right side.

80%

Graph it on Desmos

Most are read by graphing and pulling the center and radius off the picture.

the hazard

General form

Completing the square is where a sign or halving slips; graphing skips it.

How to recognize circle equations questions

An equation has both x squared and y squared with equal coefficients, or the word circle appears.
You are asked for the center, the radius, the diameter, or a point on the circle.
Standard form is given, or a general form to convert.
Choices are points, lengths, or equations.

Why students miss these

The sign flip and the squared radius are what trip people. The center of (x + 3) squared plus (y - 6) squared is negative 3 and positive 6, the opposite of what shows, and a right side of 64 is a radius of 8, not 64. General-form equations add a completing-the-square step where a sign or a halving slips. Graphing in Desmos draws the circle and sidesteps all of it.

The step-by-step method

1
Reach standard form
Aim for (x - h) squared plus (y - k) squared equals r squared; if the equation is expanded, complete the square in x and y.
2
Read the center, signs flipped
The center is (h, k), opposite the signs shown: (x + 3) gives an x of negative 3.
3
Square-root for the radius
The right side is r squared, so the radius is its square root; an 81 gives 9.
4
Answer the exact quantity
Center, radius, or diameter; the diameter is twice the radius.

Solving it on Desmos

Type the equation. Enter it in standard or general form and Desmos draws the circle.
Read center and radius. See the center and click a point on the circle; the distance to the edge is the radius.
Answer the asked value. Take the center, radius, or diameter from the graph.

Full Desmos walkthrough for circle equations→

When to use it: 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.

See it live: open a real question in the same Desmos calculator you get on Perfect1600, with the equations already typed in.

Worked examples

Easy example

The graph of

(x - 6)^{2} + (y - 3)^{2} = 81

is a circle. What is the radius of this circle?

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. $r^{2} = 81$ , so $r = \sqrt{81} = 9$ .

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

Explanation

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

Easy example

The graph of

x^{2} + y^{2} = 36

is a circle. What is the radius of this circle?

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

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

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

D: Incorrect. This triples the radius.

Explanation

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

Easy example

A circle in the

xy

-plane has equation

(x+8)^2+(y+2)^2=144

. Which of the following points lies on this circle?

(3,-2)

(4,-3)

(4,-2)

(4,0)

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

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

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

D: Incorrect. This uses $y=0$ instead of the center's $y$ -coordinate: $(4,0)$ 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.

The common traps

Pattern	What it does	The tell
Center sign not flipped	Read the center with the same signs shown in the equation.	The center's coordinates take the opposite sign of the numbers inside.
Radius versus radius squared	Reported the right side as the radius.	The right side is r squared; the radius is its square root.
Completing-the-square slip	Halved or signed a term wrong when converting general form.	Add the same amount to both sides, halve before squaring, or just graph.
Radius versus diameter	Gave the radius when the diameter was asked.	The diameter is twice the radius.

Try it: two real questions

Question 1easy

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

Question 2easy

The graph of $(x - 2)^{2} + (y - 5)^{2} = 49$ 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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Often confused with

Tangent Lines to a Circle

Common questions

How do you find the center and radius of a circle on the SAT?

Put it in standard form, (x - h) squared plus (y - k) squared equals r squared. The center is (h, k) with signs flipped, and the radius is the square root of the right side. Or graph it in Desmos.

Why is the center's sign opposite the equation?

Because standard form subtracts the center: (x - h). So (x + 3) is really (x - negative 3), putting the center's x at negative 3, the opposite of the 3 shown.

How do I convert a circle from general form to standard form?

Complete the square separately in x and y: halve each linear coefficient, square it, and add it to both sides. Or skip this by graphing in Desmos.

Can Desmos graph a circle equation?

Yes. Type it in any form and Desmos draws the circle; you can see the center and measure the radius, avoiding the sign-flip and completing-the-square mistakes.

How do I find the radius if the equation gives r squared?

Take the square root of the right side. An 81 gives a radius of 9, a 100 gives 10, and the diameter is twice that.

Practice circle equations the way it is tested

Start with the free 16-question diagnostic, then drill this type with step-by-step reasoning on every question.

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What the question bank shows

How to recognize circle equations questions

Why students miss these

The step-by-step method

Reach standard form

Read the center, signs flipped

Square-root for the radius

Answer the exact quantity

Solving it on Desmos

Worked examples

The common traps

Try it: two real questions

Question 3 is ready when you are

Often confused with

Related reading

Common questions

How do you find the center and radius of a circle on the SAT?

Why is the center's sign opposite the equation?

How do I convert a circle from general form to standard form?

Can Desmos graph a circle equation?

How do I find the radius if the equation gives r squared?

Practice circle equations the way it is tested