Circles: SAT Practice Questions & Study Guide
Questions covering circle area, circumference, arc length, sector area, central and inscribed angles, and the equation of a circle in the coordinate plane.
Understanding Circles on the SAT
Circle problems on the SAT cluster around two key formulas—area A = πr² and circumference C = 2πr—and the proportion that relates a central angle to arc length and sector area. That proportion is: (central angle θ in degrees) / 360° = (arc length) / (circumference) = (sector area) / (total area). Write this proportion out at the start of any arc or sector problem, fill in what you know, and solve for the unknown. This single structure handles nearly every non-coordinate circle question.
The central angle theorem states that a central angle (vertex at the center) is equal to the arc it intercepts. The inscribed angle theorem states that an inscribed angle (vertex on the circle's circumference) equals half the central angle that intercepts the same arc, so inscribed angle = (intercepted arc) / 2. When an inscribed angle intercepts a semicircle (a diameter), the inscribed angle is always 90°—this is a frequently tested special case. Two inscribed angles that intercept the same arc are equal to each other.
A tangent line to a circle is perpendicular to the radius drawn to the point of tangency. This creates a right angle, allowing you to apply the Pythagorean theorem. Problems often give the length of a tangent segment from an external point and the radius, and ask for the distance from the external point to the center.
In the coordinate plane, the equation of a circle with center (h, k) and radius r is (x − h)² + (y − k)² = r². The SAT sometimes gives this equation in expanded form and asks you to identify the center or radius by completing the square. Remember that the signs are reversed: center (3, −5) gives (x − 3)² + (y + 5)² = r².
Key Rules & Formulas
Memorize these rules — they come up directly in SAT questions.
Arc/sector proportion: θ/360 = arc length / (2πr) = sector area / (πr²)
Circle with r = 6, central angle = 120°: arc length = (120/360)(2π·6) = 4π
Inscribed angle = half the intercepted arc (= half the central angle for the same arc)
Inscribed angle intercepting a 100° arc = 50°
Inscribed angle in a semicircle = 90°
Any triangle inscribed in a circle where one side is a diameter contains a 90° angle at the third vertex
Tangent–radius perpendicularity: radius to point of tangency ⊥ tangent line
External point 13 units from center, radius = 5: tangent length = √(13² − 5²) = √144 = 12
Circle equation: (x − h)² + (y − k)² = r²; center (h, k), radius r
(x − 2)² + (y + 3)² = 25 has center (2, −3) and radius 5
Circles Practice Questions
Select an answer and click Check Answer to reveal the full explanation. Questions go from easiest to hardest.
A circle has a radius of 9 cm. What is its circumference in terms of π?
A circle has a central angle of 90° and a radius of 8 cm. What is the arc length intercepted by this central angle in terms of π?
The equation of a circle is (x − 3)² + (y + 4)² = 25. What is the radius of the circle?
A circle with radius 10 cm has a central angle of 120°. What is the area of the sector formed by this central angle in terms of π?
In a circle, an inscribed angle intercepts an arc of 84°. What is the measure of the inscribed angle?
A point P is located 17 cm from the center of a circle with radius 8 cm. A tangent is drawn from point P to the circle. What is the length of the tangent segment from P to the point of tangency?
A circle has the equation x² + y² − 6x + 8y − 11 = 0. What is the center of the circle?
In a circle with center O and radius 13 cm, chord AB is drawn such that the perpendicular from the center to the chord bisects it. If the distance from the center to the chord is 5 cm, what is the length of chord AB?
In a circle, central angle AOB = 140°. Point C is on the circle on the major arc (the larger arc from A to B not directly spanned by the central angle). What is the measure of inscribed angle ACB?
A sector of a circle has an arc length of 6π cm and the sector's area is 27π cm². What is the radius of the circle?
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Common Mistakes to Avoid
These are the most frequent errors students make on Circles questions. Knowing them in advance prevents costly point losses.
- !Using diameter instead of radius in the area or circumference formula
- !Setting up the arc proportion incorrectly by confusing arc length with arc measure (degrees)
- !Forgetting to halve the intercepted arc when computing an inscribed angle
- !In the coordinate circle equation, misreading the sign: (x − 3)² gives center x = 3, not x = −3
- !Not recognizing that an angle inscribed in a semicircle is always 90°, leading to unnecessary algebra
SAT Strategy Tips: Circles
Write the proportion θ/360 = arc/circumference = sector/area at the top of every circle problem involving arc or sector—fill in two known values and solve for the third
For inscribed angle problems, always ask: is the vertex at the center (central angle) or on the circle (inscribed angle)? The answer tells you whether to use the arc directly or halve it
For circle equation problems, put the equation in standard form first by completing the square if needed—don't try to identify center and radius from an expanded form
When a tangent and radius are given, immediately draw the right angle at the point of tangency and apply the Pythagorean theorem
Other Geometry & Trigonometry Subtopics
Area and Volume
Questions asking you to compute or compare areas of 2D figures and surface areas and volumes of 3D solids.
Lines, Angles, and Triangles
Questions covering angle relationships formed by parallel lines, properties of triangles, congruence, similarity, and the triangle inequality.
Right Triangles and Trigonometry
Questions applying the Pythagorean theorem, SOH-CAH-TOA, and complementary angle trig identities to find side lengths and angle measures in right triangles.
Master Circles on the SAT
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