What circle equation forms do I need to know?

Standard form (x − h)² + (y − k)² = r² with center (h, k) and radius r, and the expanded form the SAT gives when it wants you to complete the square. Moving between the two is the core skill.

When do I use s = rθ?

Only when θ is in radians. With degrees, use the fraction of the circle: arc length is (central angle ÷ 360°) × 2πr and sector area is (central angle ÷ 360°) × πr². Same idea either way, the angle measures how much of the circle you keep.

How are inscribed and central angles related?

An inscribed angle is half the central angle that opens onto the same arc. So an arc of 80° gives a central angle of 80° and an inscribed angle of 40°, and a diameter always inscribes a 90° angle.

What does a tangent line guarantee?

A tangent is perpendicular to the radius at the point of tangency. The SAT uses that hidden right angle to smuggle a Pythagorean theorem step into a circle problem, so draw the radius to the tangent point first.

Can Desmos find a circle's center and radius?

Graphically, yes. Paste the expanded equation in and Desmos draws the circle, so you can read off the center and radius and confirm your completed square. When the answer choices are symbolic you still need the algebra itself.

Geometry and Trig · Skill 4 of 4

SAT Circles Practice

Circle questions on the digital SAT split into two families: equation questions that want you to complete the square, and figure questions about arcs, sectors, and tangents. Both run on a small set of exact relationships, and both are very learnable with the right repetitions.

Domain: Geometry and Trig
About 15% of the test is Geometry and Trig
Difficulty: easy to hard
Free, no account

Drill circles now → See all Geometry and Trig skills

The patterns the SAT actually uses

Four patterns carry every circle question. Two live in the coordinate plane, two live in the figure, and all four reward exactness.

Pattern 01

Read the standard form

(x − h)² + (y − k)² = r² hands you center (h, k) and radius r. Watch the signs: (y + 2)² means k = −2, not 2.

Pattern 02

Complete the square

The expanded equation hides the center. Group the x terms and y terms, add both completing constants to both sides, and the standard form reappears.

Pattern 03

Slice arcs and sectors

The central angle fixes the fraction of the circle: arc length is (angle ÷ 360°) × 2πr, sector area is (angle ÷ 360°) × πr², and in radians s = rθ.

Pattern 04

Exploit the tangent

A tangent line is perpendicular to the radius at the touch point. That hidden right angle turns tangent questions into Pythagorean theorem questions.

One worked example, start to finish

Worked example · medium

The equation x² + y² − 6x + 4y − 12 = 0 defines a circle in the xy plane. What is the radius of the circle?

Group the variables and move the constant: (x² − 6x) + (y² + 4y) = 12.
Complete each square: half of 6 is 3 and 3² = 9; half of 4 is 2 and 2² = 4. Add 9 and 4 to both sides.
Rewrite: (x − 3)² + (y + 2)² = 12 + 9 + 4 = 25. The center is (3, −2) and r² = 25.
So r = √25 = 5. Check by expanding: x² − 6x + 9 + y² + 4y + 4 = x² + y² − 6x + 4y + 13, and setting that equal to 25 gives back x² + y² − 6x + 4y − 12 = 0 exactly.

Answer: radius = 5

New constants each time this regenerates, same ritual: group, halve, square, add to both sides, read the center and radius.

Where students lose the point

Flipping the center's signs. (x − 3)² + (y + 2)² = 25 has center (3, −2), not (−3, 2). The equation stores each coordinate with its sign reversed, and the flipped center is always a choice.
Adding constants to one side only. Completing the square added 9 and 4 on the left, so the right side must grow by 13 too. Forgetting that shrinks the radius and matches a planted choice.
Reporting r² instead of r. The equation ends in 25; the radius is 5. Both numbers appear in the choices every single time this question is asked.
Using s = rθ with degrees. That formula needs radians. With degrees, take the fraction of the circle: (angle ÷ 360°) × 2πr for arc length, and the same fraction of πr² for sector area.

Using Desmos here

Complete the square by hand, then let Desmos referee: paste x² + y² − 6x + 4y − 12 = 0 straight into the graphing window and the circle appears. If its center sits at (3, −2) and it passes through (8, −2), your radius of 5 is confirmed. The check costs ten seconds and catches every sign slip, which is exactly the error this question type is designed to harvest.

Why drilling here is different

Circle drills on Satified rebuild themselves on every load: new equations to unpack, new arcs and tangents to slice, at easy, medium, and hard levels. The completing the square ritual gets real repetition with numbers that never repeat, and every answer across the 1,483 question bank has been independently verified.

A new circle, every single time.

Start this skill free →

Questions students ask

What circle equation forms do I need to know?: Standard form (x − h)² + (y − k)² = r² with center (h, k) and radius r, and the expanded form the SAT gives when it wants you to complete the square. Moving between the two is the core skill.
When do I use s = rθ?: Only when θ is in radians. With degrees, use the fraction of the circle: arc length is (central angle ÷ 360°) × 2πr and sector area is (central angle ÷ 360°) × πr². Same idea either way, the angle measures how much of the circle you keep.
How are inscribed and central angles related?: An inscribed angle is half the central angle that opens onto the same arc. So an arc of 80° gives a central angle of 80° and an inscribed angle of 40°, and a diameter always inscribes a 90° angle.
What does a tangent line guarantee?: A tangent is perpendicular to the radius at the point of tangency. The SAT uses that hidden right angle to smuggle a Pythagorean theorem step into a circle problem, so draw the radius to the tangent point first.
Can Desmos find a circle's center and radius?: Graphically, yes. Paste the expanded equation in and Desmos draws the circle, so you can read off the center and radius and confirm your completed square. When the answer choices are symbolic you still need the algebra itself.

Keep going

Circles close out the domain, and they borrow from every other geometry skill on this list.