Math, Geometry and Trigonometry, Stage 3
Circles on the SAT: lesson and practice
Arc length and sector area are fractions of the whole circle: multiply the circumference or area by angle ÷ 360°. The equation (x − h)² + (y − k)² = r² has center (h, k) and radius r; the number on the right is r², not r. In radians, arc length = rθ.
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| Find | Formula |
|---|---|
| Arc length (degrees) | (θ/360) × 2πr |
| Sector area | (θ/360) × πr² |
| Arc length (radians) | rθ |
| Center and radius | (x + 3)² + (y − 2)² = 25 → (−3, 2), r = 5 |
Completing the square
x² + 12x + y² − 6y + 41 = 0
(x + 6)² + (y − 3)² = 36 + 9 − 41 = 4
radius 2
(x + 6)² + (y − 3)² = 36 + 9 − 41 = 4
radius 2
Common mistakes
Center signsFlip the signs inside the brackets.
Using r² as rTake the square root of the right side.
Arc vs sectorArcs use circumference; sectors use area.
Sample questions with answers
1. Radius 10, central angle 120°. Arc length?
- 20π/3 (correct)
- 40π/3
- 100π/3
- 20π
(120/360) × 20π = 20π/3.
2. Center of (x − 8)² + (y − 3)² = 9?
- (−8, −3)
- (8, 3) (correct)
- (8, −3)
- (−8, 3)
Flip the signs: (8, 3).
3. Radius of (x − 1)² + (y + 4)² = 36?
- 36
- 18
- 6 (correct)
- 4
r² = 36, so r = 6.
Frequently asked questions
What circle topics are on the SAT?
Arcs, sectors, circle equations and radians. Geometry and Trigonometry makes up about 15% of SAT Math, roughly 5 to 7 questions. A reference sheet with key formulas is available during the test.
How do I find the radius from a circle equation?
Take the square root of the number on the right side of (x − h)² + (y − k)² = r².
What is arc length in radians?
Arc length = radius × angle in radians.