Digital SAT · Math
Circles on the Digital SAT: All Question Types
Circles covers the equation of a circle in the coordinate plane and the geometry of arcs, sectors, and angles. Equation questions often require completing the square to find the center and radius from an expanded form. Arc length and sector area are fractions of the circumference and total area, set by the central angle over 360 degrees, and radian measure expresses that fraction directly. Tangent-line questions use the fact that a tangent meets the radius at a right angle. This is a Geometry and Trigonometry skill and one of the more specialized. The recurring setup is turning a central angle into a fraction of the whole circle, so identifying that fraction first makes arc and sector questions a single multiplication.
College Board skill: Geometry and Trigonometry: Circles
How Circles is tested
- how often it appears
- ~2 per test how often it appears
- typical difficulty
- Medium to hard typical difficulty
- practice questions in our bank
- 198 practice questions in our bank
Equation and proportions
Use (x-h)^2 + (y-k)^2 = r^2 for the circle, and set arc length and sector area as proportions of the central angle.
Frequency and difficulty come from this skill's questions across our assembled full-length Digital SAT forms; the practice count is how many drills of these types are in our bank.
4 question types in Circles
An arc is a slice of the circle's edge, and a sector is a slice of its inside, both sized by the central angle.
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.
Degrees and radians are two scales for the same angle.
A tangent line touches a circle at exactly one point, and at that point it is perpendicular to the radius.
Common questions
How do I find the center and radius of a circle?
From the standard equation, the center is the point that makes each squared term zero and the radius is the square root of the constant. If the equation is expanded, complete the square first to reach that form.
How do I find arc length or sector area?
Take the central angle as a fraction of the whole circle, the angle over 360 degrees, and multiply that fraction by the circumference for arc length or by the total area for sector area.
What is special about a tangent line?
A tangent touches the circle at one point and is perpendicular to the radius drawn to that point. That right angle sets up a right triangle, which most tangent-line questions rely on.
Practice Circles the way it is tested
Start with the free 16-question diagnostic, then drill any of these types with step-by-step reasoning.