Circles: Central & Inscribed Angles
Master every angle-arc relationship tested on the SAT Math section
SAT Geometry • Circle Theorems • Angle-Arc Relationships
1. Introduction & Key Vocabulary
Core Definitions
| Term | Definition |
|---|---|
| Central Angle | An angle whose vertex is at the center of the circle; its sides are radii. |
| Inscribed Angle | An angle whose vertex is on the circle and whose sides are chords. |
| Intercepted Arc | The arc that lies in the interior of an angle and has its endpoints on the angle's sides. |
| Arc Measure | Measured in degrees; equals the central angle that intercepts it. |
| Semicircle | An arc of 180°; half the circle, cut by a diameter. |
| Cyclic Quadrilateral | A quadrilateral with all four vertices on a circle. |
Why These Angles Matter
- They form the foundation of all circle-angle problems on the SAT.
- Understanding the relationship between an angle and its intercepted arc lets you solve problems quickly.
- Many SAT questions combine these with algebra — setting up equations with $x$.
- Recognizing a 90° inscribed angle instantly tells you a diameter is present.
SAT Tip: Almost every circle-angle SAT question boils down to one idea: where is the vertex? That determines which formula to use.
Interactive: Central Angle vs. Inscribed Angle
Interactive visualization — available in the full lesson
Drag point P around the circle. Notice the inscribed angle is always half the central angle!
2. Central Angles
Definition & Core Rule
A central angle is formed by two radii with its vertex at the center of the circle.
$$\text{Central Angle} = \text{Intercepted Arc}$$
Key Facts
- A central angle of $60°$ intercepts an arc of $60°$.
- A central angle of $90°$ intercepts an arc of $90°$ (quarter circle).
- A central angle of $120°$ intercepts an arc of $120°$.
- A central angle of $180°$ intercepts a semicircle.
- A full rotation: central angle of $360°$ = full circle.
- All central angles around the center sum to $360°$.
- In the same circle, congruent central angles intercept congruent arcs.
Interactive visualization — available in the full lesson
Quarter Circle
$90° \to 90°$ arc
Semicircle
$180° \to 180°$ arc
Full Circle
$360° \to 360°$ arc