🎯 Learning Objectives
- All the parts of a circle: radius, diameter, chord, arc, sector.
- Area and circumference formulas, plus how arc length and sector area relate.
- Radian measure: the SAT version simplified.
- Equation of a circle in the xy-plane and how to spot the center and radius.
- 10 SAT-style practice questions.
1. Anatomy of a Circle
Part | Definition | Symbol |
Center | Point at the middle | O |
Radius | Center to edge | r |
Diameter | Edge to edge through center | d = 2r |
Chord | Edge to edge (any straight segment) | : |
Arc | Curved part of the circle | : |
Sector | Pie-slice region (two radii + arc) | : |
Tangent | Touches the circle at exactly one point | : |
2. The Two Big Formulas
Distance around a full circle
Circumference = 2π × r = π × d
Region inside a full circle
Area = π × r²
3. Arc Length and Sector Area: The Fraction Strategy
An arc and a sector are FRACTIONS of the full circle. The fraction is the same as the central angle divided by 360°.
Arc length = (θ / 360°) × 2π × r
Sector area = (θ / 360°) × π × r²
⚡ Strategy: Memorize the Fractions
- 60° → 1/6 of the circle
- 90° → 1/4 of the circle
- 120° → 1/3 of the circle
- 180° → 1/2 of the circle
Then arc length = fraction × circumference, sector area = fraction × area.
4. Radian Measure
A radian is just another unit for angles. Instead of dividing the circle into 360 equal pieces, we divide it into 2π equal pieces.
The conversion you must memorize
180° = π radians
Common Conversions
Degrees | Radians |
30° | π/6 |
45° | π/4 |
60° | π/3 |
90° | π/2 |
180° | π |
360° | 2π |
To convert: degrees × (π / 180°) → radians
To convert: radians × (180° / π) → degrees
💡 Pro Tip: Arc Length in Radians
If the angle is in radians, arc length simplifies to s = r·θ. No 360° fraction needed.
Example: A circle of radius 5 has an arc subtending angle π/3 radians. Arc length = 5 × π/3 = 5π/3.
5. Equation of a Circle in the xy-Plane
Center (h, k); radius r
(x − h)² + (y − k)² = r²
To read the center, take the OPPOSITE sign of what you see in the parentheses. (x − 3)² has center x = 3, but (x + 3)² has center x = −3.
⚠️ Common Mistakes
The right side is r², not r. If the equation says (x − 2)² + (y − 5)² = 16, the radius is √16 = 4, not 16.
6. Tangent Lines
A tangent line touches the circle at exactly one point. At that point, the tangent is PERPENDICULAR to the radius. This perpendicularity is the key to most tangent-line problems.
7. Strategies
⚡ For arc/sector questions, find the fraction first
θ / 360° (or θ / 2π in radians).
⚡ Memorize 30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2
Lets you skip the conversion every time.
⚡ For circle equations, take opposite signs for center
(x + 3)² has center at x = −3.
⚡ Tangent ⊥ radius at the point of contact
Creates an automatic right triangle to work with.
⚡ For inscribed angles, use the inscribed angle theorem
An inscribed angle is HALF the central angle that subtends the same arc.
8. Summary
📌 Key Takeaways
- Circumference: 2πr.
- Area: πr².
- Arc / Sector: fraction (θ/360°) of circle.
- Radians: 180° = π.
- Equation: (x − h)² + (y − k)² = r².
- Tangent ⊥ radius at point of contact.
9. Practice: 10 Questions
Q1. A circle has radius 7. Circumference?
A) 7π B) 14π C) 49π D) 21π
Hint: C = 2πr.
Q2. A circle has radius 6. Area?
A) 12π B) 36π C) 72π D) 18π
Hint: A = πr².
Q3. A sector with central angle 90° has radius 4. Sector area?
A) 4π B) 8π C) 16π D) 2π
Hint: (1/4) × π × 16.
Q4. What is the arc length of a 60° arc on a circle of radius 12?
A) 4π B) 6π C) 12π D) 2π
Hint: (1/6) × 2π × 12.
Q5. Convert 90° to radians.
A) π/2 B) π/3 C) π/4 D) π/6
Hint: Memorized: 90° = π/2.
Q6. The equation of a circle is (x − 2)² + (y + 5)² = 49. Center?
A) (2, 5) B) (−2, −5) C) (2, −5) D) (−2, 5)
Hint: Opposite signs of what's shown.
Q7. For the same circle, what is the radius?
A) 7 B) 14 C) 49 D) √7
Hint: r² = 49 → r = 7.
Q8. A tangent at point P meets a radius. The angle between them is…
A) 0° B) 45° C) 90° D) 180°
Hint: Tangent ⊥ radius.
Q9. A 120° sector of a circle of radius 6 has area…
A) 12π B) 36π C) 8π D) 4π
Hint: (120/360) × 36π = 12π.
Q10. The arc length subtending angle π/4 radians on a circle of radius 8 is…
A) π B) 2π C) 4π D) 8π
Hint: s = rθ = 8 × π/4 = 2π.
Answer Key & Worked Solutions
# | Answer | Reasoning |
Q1. | B) 14π | 2π × 7. |
Q2. | B) 36π | π × 36. |
Q3. | A) 4π | (1/4) × 16π. |
Q4. | A) 4π | (1/6) × 24π. |
Q5. | A) π/2 | Standard conversion. |
Q6. | C) (2, −5) | Opposite signs of (x − 2), (y + 5). |
Q7. | A) 7 | √49. |
Q8. | C) 90° | Tangent ⊥ radius. |
Q9. | A) 12π | (1/3) × 36π. |
Q10. | B) 2π | s = r·θ = 8 × π/4. |