SAT Math skill page

SAT Circle Equations Practice

Connect circle formulas with coordinate geometry so radius, center, and distance questions become easier to spot.

Practice circle equations in Dolphin Practice questions

10-15 min practice time3 questions on pageGeometry and Trigonometry

Practice time10-15 min

Question bank3 questions

Best forGeometry and Trigonometry

What this tests

What to know for this SAT skill

Use center-radius form of a circle equation
Find radius from diameter, area, or coordinate distance
Apply circumference and area formulas
Recognize when a circle question is really a distance question

Dolphin question bank

Practice your skill

Question 1 of 3Hard

A circle in the xy-plane has equation (x+3)^2 + (y-1)^2 = 25. Which of the following points does NOT lie in the interior of the circle?

(-7, 3)
(-3, 1)
(0, 0)
(3, 2)

Show answer and solution

Correct answerD. (3, 2)

Solution walkthrough

The circle with equation $(x + 3)^{2} + (y - 1)^{2} = 25$ has center $(- 3, 1)$ and radius 5.
For a point to be inside of the circle, the distance from that point to the center must be less than the radius, 5.
The distance between $(3, 2)$ and $(- 3, 1)$ is $(- 3 - 3)^{2} + (1 - 2)^{2} = (- 6)^{2} + (- 1)^{2} = 37$ , which is greater than 5.
Therefore, $(3, 2)$ does NOT lie in the interior of the circle.

Why the other choices miss

Choice A is incorrect. The distance between

(- 7, 3)

and

(- 3, 1)

(- 7 + 3)^{2} + (3 - 1)^{2} = (- 4)^{2} + (2)^{2} = 20

, which is less than 5, and therefore

(- 7, 3)

lies in the interior of the circle. Choice B is incorrect because it is the center of the circle. Choice C is incorrect because the distance between

(0, 0)

and

(- 3, 1)

(0 + 3)^{2} + (0 - 1)^{2} = (3)^{2} + (1)^{2} = 10

, which is less than 5, and therefore

(0, 0)

is in the interior of the circle.

Question 2 of 3Hard

A circle in the xy-plane has its center at (-5, 2) and has a radius of 9. An equation of this circle is

x^{2}

y^{2}

+ ax + by + c = 0, where a, b, and c are constants. What is the value of c?

Show answer and solution

Correct answerC. -52

Solution walkthrough

The correct answer is $- 52$ . The equation of a circle in the xy-plane with its center at $(h, k)$ and a radius of $r$ can be written in the form $(x - h)^{2} + (y - k)^{2} = r^{2}$ . It's given that a circle in the xy-plane has its center at $(- 5, 2)$ and has a radius of 9.
Substituting $- 5$ for $h$ , 2 for $k$ , and 9 for $r$ in the equation $(x - h)^{2} + (y - k)^{2} = r^{2}$ yields $(x - (- 5))^{2} + (y - 2)^{2} = 9^{2}$ , or $(x + 5)^{2} + (y - 2)^{2} = 81$ . It's also given that an equation of this circle is $x^{2} + y^{2} + a x + b y + c = 0$ , where $a$ , $b$ , and $c$ are constants. Therefore, $(x + 5)^{2} + (y - 2)^{2} = 81$ can be rewritten in the form $x^{2} + y^{2} + a x + b y + c = 0$ .
The equation $(x + 5)^{2} + (y - 2)^{2} = 81$ , or $(x + 5) (x + 5) + (y - 2) (y - 2) = 81$ , can be rewritten as $x^{2} + 5 x + 5 x + 25 + y^{2} - 2 y - 2 y + 4 = 81$ . Combining like terms on the left-hand side of this equation yields $x^{2} + y^{2} + 10 x - 4 y + 29 = 81$ . Subtracting 81 from both sides of this equation yields $x^{2} + y^{2} + 10 x - 4 y - 52 = 0$ , which is equivalent to $x^{2} + y^{2} + 10 x + (- 4) y + (- 52) = 0$ .
This equation is in the form $x^{2} + y^{2} + a x + b y + c = 0$ . Therefore, the value of $c$ is $- 52$ .

Question 3 of 3Hard

The equation (x + 6)^2 + (y + 3)^2 = 121 defines a circle in the xy-plane. What is the radius of the circle?

Show answer and solution

Correct answerC. 11

Solution walkthrough

A circle with equation $(x - a)^{2} + (y - b)^{2} = r^{2}$ , where $a$ , $b$ , and $r$ are constants, has center $(a, b)$ and radius $r$ .
Therefore, the radius of the given circle is $121$ , or 11.

Avoid these traps

Common mistakes on this skill

Confusing radius and diameter

Diameter is twice the radius. Many SAT answer choices test this swap.

Reading r squared as r

In (x - h)^2 + (y - k)^2 = r^2, the number on the right is r squared.

Missing sign changes in the center

The center of (x - 3)^2 + (y + 2)^2 is (3, -2), not (-3, 2).

Study plan

How to practice this skill in Dolphin

Memorize area, circumference, and center-radius form.
Mark whether the problem asks for radius, diameter, area, or circumference.
For coordinate problems, draw the center and point before using distance.
Check if the final answer should be exact, decimal, or in terms of pi.

Practice circle equations in Dolphin

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FAQ

Questions about SAT Circle Equations Practice

Are circle equations on SAT Math?

Yes. Circle questions can ask about formulas, coordinate geometry, radius, diameter, or interpreting equation form.

What circle equation should I know?

Know (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center and r is the radius.

How do circle questions connect to distance?

A radius is the distance from the center to a point on the circle, so coordinate circle problems often use the distance formula.