PSAT/NMSQT Math: Circles
Circle questions on the PSAT/NMSQT test your comfort with several connected ideas: basic measurements like circumference and area, angle-based measurements like arc length and sector area, and the algebraic equation of a circle. This free, complete lesson covers circumference and area, arc length and sector area, radians and central angles, and the equation of a circle. Each idea is followed by a fully worked example, and then a set of original practice problems with instant feedback and full explanations. Everything here is free, and you can keep practicing afterward with the linked quiz below.
1. Circumference and Area
The diameter of a circle is twice its radius: d = 2r. The circumference (the distance around the circle) is C = 2πr, and the area is A = πr².
Worked Example: A circle has a radius of 9. Find its circumference and its area, in terms of π.
Circumference = 2πr = 2π(9) = 18π.
Area = πr² = π(9)² = 81π
1. A circle has a radius of 6. Find its diameter.
Explanation: Diameter = 2 × radius = 2 × 6 = 12.
2. A circle has a radius of 9. Find its circumference, in terms of π. (see worked example above)
Explanation: Circumference = 2πr = 2π(9) = 18π.
3. A circle has a radius of 9. Find its area, in terms of π. (see worked example above)
Explanation: Area = πr² = π(9)² = 81π.
2. Arc Length and Sector Area
An arc is a portion of a circle's circumference, and a sector is a pie-slice-shaped portion of a circle's area. Both are found by taking the fraction of the circle covered by a central angle, out of 360°, and multiplying by the full circumference or full area.
Worked Example: A circle has a radius of 12. Find the length of an arc cut off by a central angle of 60°, in terms of π.
The fraction of the circle is 60/360 = 1/6.
Arc length = (1/6) × 2π(12) = (1/6) × 24π = 4π
1. A circle has a radius of 12. Find the length of an arc cut off by a central angle of 60°, in terms of π. (see worked example above)
Explanation: Fraction of circle = 60/360 = 1/6. Arc length = (1/6) × 2π(12) = (1/6) × 24π = 4π.
2. A circle has a radius of 10. Find the area of a sector with a central angle of 90°, in terms of π.
Explanation: Fraction of circle = 90/360 = 1/4. Full area = π(10)² = 100π. Sector area = (1/4) × 100π = 25π.
3. A circle has a circumference of 36π. An arc on this circle has a length of 6π. What is the measure of the arc's central angle?
Explanation: The arc is 6π/36π = 1/6 of the full circle. 1/6 of 360° = 60°.
3. Radians and Central Angles
Radians are another unit for measuring angles, where a full circle equals 2π radians (instead of 360°). To convert between degrees and radians, use the relationship 180° = π radians.
Worked Example: Find the radian measure of a central angle that corresponds to 1/4 of a full circle.
(1/4) × 2π = π/2 radians
1. Find the radian measure of a central angle that corresponds to 1/4 of a full circle. (see worked example above)
Explanation: A full circle is 2π radians. One-fourth of that is (1/4) × 2π = π/2.
2. Convert a central angle of 120° to radians.
Explanation: Multiply by π/180: 120 × π/180 = 2π/3.
3. An angle measures 5π/6 radians. Convert this to degrees.
Explanation: Multiply by 180/π: (5π/6) × (180/π) = (5 × 180)/6 = 150°.
4. The Equation of a Circle
The standard form equation of a circle with center (h, k) and radius r is (x − h)² + (y − k)² = r². The center's coordinates are the values subtracted from x and y (watch the sign flip), and the radius is the square root of the number on the right side.
Worked Example: Find the center and radius of the circle (x + 3)² + (y − 2)² = 49.
Rewrite as (x − (−3))² + (y − 2)² = 49, so center = (−3, 2).
Radius = √49 = 7
1. Find the center and radius of the circle (x + 3)² + (y − 2)² = 49. (see worked example above)
Explanation: The equation form is (x − h)² + (y − k)² = r². Here h = −3 and k = 2, so the center is (−3, 2), and r = √49 = 7.
2. Write the standard form equation of a circle with center (4, −1) and radius 5.
Explanation: Substitute h = 4, k = −1, r = 5 into (x − h)² + (y − k)² = r²: (x − 4)² + (y − (−1))² = 5², which is (x − 4)² + (y + 1)² = 25.
3. The circle (x − 2)² + (y − 5)² = 9 is shifted 1 unit left, 3 units up, and its radius is doubled. Write the new equation.
Explanation: The original center (2, 5) shifts 1 left and 3 up to (1, 8). The original radius of √9 = 3 doubles to 6, so r² = 36. New equation: (x − 1)² + (y − 8)² = 36.
Common Mistakes to Avoid
- Confusing radius and diameter when plugging into the circumference or area formula, always double-check which one is given before substituting.
- Forgetting to convert a central angle into a fraction of 360° (or 2π radians) before applying it to arc length or sector area, these formulas always start from that fraction.
- Mixing up the degree-to-radian and radian-to-degree conversion factors. Multiply by π/180 to go from degrees to radians, and by 180/π to go the other way.
- Misreading the sign of the center's coordinates in the circle equation. Since the formula subtracts h and k, a "+3" inside the parentheses actually means h = −3.
- Forgetting to square root the right-hand side of the circle equation to find the radius, the number there is r², not r itself.
Frequently Asked Questions
Why do radians matter if degrees work fine?
Radians connect an angle directly to arc length and are the standard unit in more advanced math, including trigonometric functions and calculus. The PSAT/NMSQT expects you to move comfortably between both systems.
How do I remember the circle equation's sign convention?
Since the formula is (x − h)² + (y − k)² = r², rewrite any "+" inside the parentheses as "minus a negative" first, like rewriting (x + 3)² as (x − (−3))², to clearly see that h = −3.
Where can I practice more problems like these?
The Circles quiz in the PSAT/NMSQT Math Question Bank includes additional original problems on this topic, along with quizzes covering every other Heart of Algebra, Advanced Math, Problem Solving and Data Analysis, and Geometry and Trigonometry skill tested on the PSAT/NMSQT. You can also browse the full PSAT/NMSQT Math Question Bank.