Math · Geometry and Trigonometry

Circles on the Digital SAT

Circle questions on the SAT come in two flavors. The coordinate-geometry flavor works with the standard equation (x − h)² + (y − k)² = r², asking you to read off a center and radius, complete the square to get there, write the equation from a center and radius, or shift and scale a circle to a new equation. The classic-geometry flavor works with arcs, sectors, and central angles, asking for a fraction of the circumference or the area.

Both flavors are formula-light and detail-heavy. The equation of a circle has built-in sign flips: (x − 3)² means the center is at x = 3, not x = −3, and the right side of the equation is r², not r. Arc and sector problems live or die on one idea: the central angle over 360 degrees (or over 2π radians) tells you what fraction of the whole circle you have.

Hard circle questions usually hide the standard form. You will get something like x² + y² − 6x + 4y − 12 = 0 and need to complete the square twice to expose the center and radius. The algebra is short, but skipping the step where you add the completed-square constants to the right side is the single most common way to get it wrong.

New to the Digital SAT? Start with how the test is structured and scored.

What the SAT actually tests

Reading the center (h, k) and radius r from the standard form (x − h)² + (y − k)² = r², including the sign flips
Completing the square to convert an expanded circle equation into standard form
Writing a circle's equation from its center and radius, or from its center and a point on the circle, and shifting or scaling a circle to derive the new equation
Arc length as the central angle's fraction of the circumference: (angle/360°) × 2πr
Sector area as the central angle's fraction of the circle's area: (angle/360°) × πr²
Central angles measured in both degrees and radians, and converting between them with 180° = π radians
Relationships between chords, radii, and tangent lines, such as a tangent being perpendicular to the radius at the point of tangency

Strategies that work

Flip the signs to read the center

In (x − h)² + (y − k)² = r², the center is (h, k), so the coordinates are the opposite of the signs you see. For (x − 3)² + (y + 2)² = 25, the center is (3, −2). Say it as "whatever makes each parenthesis zero" and you will never flip it backward.

Treat arcs and sectors as fractions of the circle

Arc length and sector area are both just the central angle over 360° times the whole thing: the circumference 2πr for arc length, the area πr² for sector area. Write the fraction first, then multiply. This one setup replaces every arc and sector formula.

Complete the square methodically, both variables

Group the x terms and y terms, halve each middle coefficient, square it, and add it to both sides. Whatever you add on the left to build the perfect squares must also be added on the right, and the number you end up with on the right is r², not r.

Build the equation in reverse

To write a circle's equation from a center and radius, square the radius and flip the signs of the center coordinates. If you are given a center and a point on the circle instead, the radius squared equals the sum of the squared differences between the point's coordinates and the center's coordinates. Shifting a circle changes only the center in the parentheses, and scaling the radius changes only the r² on the right.

Sketch the circle when the question turns geometric

For tangent lines, chords, or points on the circle, a quick sketch with the center and radius labeled turns an abstract question into a right-triangle problem. A tangent line is perpendicular to the radius at the point of contact, and that right angle is usually the whole solution.

Mistakes to avoid

Reading the center of (x − 3)² + (y + 2)² = 25 as (−3, 2) instead of (3, −2)
Reporting r² as the radius: if the equation equals 25, the radius is 5, not 25
Computing sector area when the question asked for arc length, or the reverse
Forgetting to add the completed-square constants to the right side when converting to standard form
Mixing degrees and radians in one calculation, such as using a radian angle over 360 instead of over 2π

Solve it with Desmos

Let the calculator do the work

The Digital SAT's built-in Desmos calculator can crack Circles questions in a few clicks. Here is the move that applies, step by step.

Find a Circle's Center and Radius from Its EquationSkip completing the square. Graph it and measure.

All 15 Desmos techniques

Practice with Nova, your Math coach

Answer right here and Nova walks you through it, just like in a drill. Difficulty labels are relative within this skill.

Tip: the Calculator button on each question opens the same Desmos graphing calculator you get on the real Digital SAT. Try solving these with it.

Question 1Easy

A circle in the xy-plane has the equation (x − 3)² + (y + 2)² = 25.

What are the center and the radius of the circle?

Pick an answer to get instant coaching from Nova.

Question 2Medium

A circle has a radius of 6, and a central angle of the circle measures 60°.

What is the length of the arc intercepted by the central angle?

Pick an answer to get instant coaching from Nova.

Question 3Hard

A circle in the xy-plane has the equation x² + y² − 6x + 4y − 12 = 0.

What is the radius of the circle?

Pick an answer to get instant coaching from Nova.

Keep missing Circles questions?

SAT Prep Quest tracks your level on this skill (and the other 27), then feeds you 5-minute drills right at your challenge zone until it is fixed.

Drill this skill free

The rest of Geometry and Trigonometry

Circles is one of 4 skills in Geometry and Trigonometry. The ideas build on each other, so it pays to learn them as a set rather than in isolation. See how Geometry and Trigonometry fits together.

Other Math skills to explore

SAT is a registered trademark of the College Board, which was not involved in producing, and does not endorse, this product. Score estimates are Learner Labs projections, not official College Board scores.