Circles: Free SAT Math Lesson
"Circles" is one of the four officially named question types in College Board's Geometry and Trigonometry domain, and it blends algebra (the equation of a circle) with geometry (arcs, sectors, and angle relationships). This free, complete lesson covers the equation of a circle and how to find its center and radius, even when you need to complete the square first, arc length and sector area using degrees, converting between degrees and radians, radian-based arc length and sector area formulas, and key circle theorems involving inscribed angles and tangent lines. Each idea is followed by a fully worked example, and then a set of original practice problems with instant feedback. Everything here is free, and you can keep practicing afterward with the full SAT Math Question Bank linked below.
1. The Equation of a Circle
The equation of a circle with center (h, k) and radius r is:
(x − h)² + (y − k)² = r²
A circle with center (h, k) and radius r
If a circle's equation is given in general (expanded) form, complete the square on the x-terms and the y-terms separately to rewrite it in center-radius form.
Worked Example: Find the center and radius of the circle x² + y² − 6x + 8y + 9 = 0.
(x² − 6x) + (y² + 8y) = −9
(x² − 6x + 9) + (y² + 8y + 16) = −9 + 9 + 16
(x − 3)² + (y + 4)² = 16
The center is (3, −4) and the radius is √16 = 4.
1. What is the center and radius of the circle (x − 2)² + (y − 7)² = 36?
Explanation: Comparing to (x − h)² + (y − k)² = r², h = 2, k = 7, and r² = 36, so r = 6.
2. What is the equation of a circle with center (−4, 1) and radius 5?
Explanation: With h = −4 and k = 1: (x − (−4))² + (y − 1)² = 5², which simplifies to (x + 4)² + (y − 1)² = 25.
3. A circle has equation x² + y² = 81. What is its radius?
Explanation: This circle is centered at the origin with r² = 81, so r = √81 = 9.
4. A circle is given by x² + y² − 6x + 8y + 9 = 0. What is the center?
Explanation: This is the worked example above: completing the square gives (x − 3)² + (y + 4)² = 16, so the center is (3, −4).
5. Using the equation from the previous problem, what is the radius?
Explanation: From (x − 3)² + (y + 4)² = 16, r² = 16, so r = 4.
2. Arc Length and Sector Area in Degrees
An arc or sector is a fraction of the whole circle, based on what fraction its central angle is of the full 360°.
arc length = θ360 × 2πr sector area = θ360 × πr²
A sector with central angle θ
Worked Example: A circle has radius 12. Find the arc length of a 60° arc.
arc length = 60360 × 2π(12) = 16 × 24π = 4π
1. A circle has radius 9. What is the arc length of a 40° arc, in terms of π?
Explanation: arc length = 40360 × 2π(9) = 19 × 18π = 2π.
2. A circle has radius 10. What is the area of a 90° sector, in terms of π?
Explanation: sector area = 90360 × π(10)² = 14 × 100π = 25π.
3. A circle has radius 6. An arc has length 2π. What is the central angle of the arc, in degrees?
Explanation: 2π = θ360 × 2π(6) = θ360 × 12π. Dividing both sides by 12π: θ360 = 16, so θ = 60°.
4. A sector of a circle with radius 8 has an area of 16π. What is the central angle of the sector, in degrees?
Explanation: 16π = θ360 × π(8)² = θ360 × 64π. So θ360 = 1664 = 14, giving θ = 90°.
5. A pizza with radius 7 inches is cut into a slice with a 45° central angle. What is the area of the slice, in terms of π?
Explanation: sector area = 45360 × π(7)² = 18 × 49π = 49π8.
3. Radians
A radian is an alternate unit for measuring angles. A full circle is 360°, which equals 2π radians. To convert:
radians = degrees × π180 degrees = radians × 180π
In radians, the arc length and sector area formulas simplify, since the fraction of the circle is just the angle divided by the full 2π:
arc length = rθ sector area = 12r²θ
Worked Example: Convert 120° to radians.
120 × π180 = 2π3
1. Convert 270° to radians.
Explanation: 270 × π180 = 270π180 = 3π2.
2. Convert 5π6 radians to degrees.
Explanation: 5π6 × 180π = 5(180)6 = 150°.
3. A circle has radius 4 and a central angle of π3 radians. What is the arc length?
Explanation: arc length = rθ = 4 × π3 = 4π3.
4. A circle has radius 6 and a central angle of π2 radians. What is the area of the sector?
Explanation: sector area = 12r²θ = 12(36)π2 = 18 × π2 = 9π.
4. Circle Theorems and Applications
Two circle relationships appear often. First, the inscribed angle theorem: an inscribed angle (with its vertex on the circle) is always half the central angle that intercepts the same arc. Second, a line tangent to a circle is always perpendicular to the radius drawn to the point of tangency.
Worked Example: A central angle intercepts an arc of 80°. What is the measure of an inscribed angle that intercepts the same arc?
inscribed angle = 80°2 = 40°
1. A central angle measures 100°. What is the measure of an inscribed angle intercepting the same arc?
Explanation: An inscribed angle is half its corresponding central angle: 100° ÷ 2 = 50°.
2. An inscribed angle measures 35°. What is the measure of the central angle intercepting the same arc?
Explanation: The central angle is double the inscribed angle: 35° × 2 = 70°.
3. An inscribed angle intercepts a semicircle (an arc of 180°). What is the measure of the inscribed angle?
Explanation: 180° ÷ 2 = 90°. This is a special case worth remembering: any inscribed angle that intercepts a semicircle is always a right angle.
4. A line is tangent to a circle at point P. What is the measure of the angle between the tangent line and the radius drawn to point P?
Explanation: A tangent line is always perpendicular to the radius at the point of tangency, so the angle between them is always 90°, regardless of the circle's size.
5. A circular track has a radius of 50 meters. A runner covers an arc with a central angle of 72°. How far did the runner run, to the nearest meter? (Use π ≈ 3.14)
Explanation: arc length = 72360 × 2π(50) = 15 × 100π = 20π ≈ 20(3.14) = 62.8, which rounds to 63 meters.
Common Mistakes to Avoid
- Reading the center coordinates with the wrong sign. In (x − h)² + (y − k)² = r², a plus sign inside the parentheses, like (x + 4)², means h = −4, not 4. Rewrite addition as subtracting a negative before identifying the center.
- Forgetting to take the square root when finding the radius. The right side of the circle equation is r², not r. If the equation ends in "= 25," the radius is 5, not 25.
- Adding the wrong constant when completing the square. When completing the square for both x and y terms, remember to add the same amount to both sides of the equation, or track it carefully as shown in the worked example.
- Mixing up the degree and radian versions of the arc length and sector area formulas. The degree version needs the 360° denominator; the radian version does not, since radians already measure angle as a fraction of the full 2π.
- Assuming an inscribed angle equals its central angle. An inscribed angle is always half of the central angle that intercepts the same arc, not equal to it.
Frequently Asked Questions
Why does the SAT test both degrees and radians for circles?
Degrees are the more intuitive, everyday unit, while radians connect angle measure directly to arc length and are essential for higher-level math and trigonometry. Being comfortable converting between the two, and knowing which version of a formula to use, is part of what's being tested.
Do I need to memorize the full formula for completing the square on a circle equation?
You should be comfortable with the process, group the x-terms together and the y-terms together, then add the square of half the coefficient to complete each square, keeping both sides of the equation balanced. It's the same technique used for completing the square on a quadratic equation, just applied twice in one problem.
What's a quick way to remember the inscribed angle theorem?
The inscribed angle is always half the central angle for the same arc. A useful special case to remember directly: any angle inscribed in a semicircle is always a right angle, 90°.
Where can I practice more problems like these?
The Circles quiz in the SAT Math Question Bank includes additional original problems on this topic, along with quizzes covering every other Algebra, Advanced Math, Problem-Solving and Data Analysis, and Geometry and Trigonometry skill on the Digital SAT.