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Circles for the PSAT/NMSQT: Complete Study Guide + Free Practice Problems
Everything the PSAT/NMSQT tests about circles in one place: circumference, arc length, area, sector area, the equation of a circle, and central and inscribed angles — with step-by-step examples, worked problems, and a free practice quiz.
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Free · No signupCircles
Circumference, arc length, area, sector area, and the equation of a circle.
Circle vocabulary
| Term | Definition |
|---|---|
| Radius | Distance from the center to any point on the circle |
| Diameter | A chord through the center; equal to 2 × radius |
| Chord | A segment connecting any two points on the circle |
| Tangent line | A line that touches the circle at exactly one point |
| Sector | A "pie slice" region bounded by two radii and an arc |
A tangent line is always perpendicular to the radius drawn to the point of tangency — this single fact unlocks many circle problems that combine circles with right triangles.
Circumference & arc length
A circle has a radius of 12. Find the length of an arc with a central angle of 90°.
Area & sector area
A circle has a radius of 8. Find the area of a sector with a central angle of 45°.
The equation of a circle
What are the center and radius of the circle (x + 2)² + (y − 4)² = 36?
In the standard form equation, the center coordinates are the values that make each term equal zero — not the numbers as they appear. (x + 2)² means h = −2, not 2.
Central & inscribed angles
Central angle
Vertex at the center of the circle. Its measure equals the measure of the arc it intercepts.
Inscribed angle
Vertex on the circle itself. Its measure is always half the measure of its intercepted arc.
A central angle intercepts an arc of 100°. What is the measure of an inscribed angle that intercepts the same arc?
An inscribed angle that intercepts a semicircle is always a right angle (90°) — a useful shortcut when a circle problem is combined with a right triangle.
Common PSAT traps
- Radius vs. diameter confusion: double-check which one is given before plugging into a formula.
- Sign errors in the equation of a circle: (x − h) means the center's x-coordinate is +h, and (x + h) means it's −h.
- Central vs. inscribed angle mix-up: a central angle equals its arc directly; an inscribed angle is half its arc.
- Forgetting to complete the square for both variables: when converting from expanded to standard form, both x and y need their own step.
Test day strategy for circles
| Question signal | Fastest approach |
|---|---|
| Arc length or sector area given a central angle | Find the angle's fraction of 360°, apply it to circumference or area |
| Equation of a circle not in standard form | Complete the square for both x and y terms |
| Inscribed angle given | Double it to find the intercepted arc, or halve the arc to find the angle |
| Angle inscribed in a semicircle | Recognize it as a 90° angle immediately |
| Tangent line problem | Draw the radius to the point of tangency — it's always perpendicular to the tangent |
Now put it to work
One comprehensive quiz covering the full topic — circumference, arc length, area, sector area, and the equation of a circle.
Circles
Circumference, arc length, area, sector area, and the equation of a circle.