A circle’s full rotation is 360°, and every arc and central angle is a slice of that whole. A central anglehas its vertex at the circle’s center, and it’s always equal to the arc it cuts off. An inscribed anglehas its vertex on the circle itself, and there’s a clean relationship between the two: an inscribed angle is always exactly half the central angle that subtends the same arc.
Let me show you one — circles
An arc measures 80°. What’s the inscribed angle that subtends it? An inscribed angle is half the central angle (which equals the arc measure here), so 80° ÷ 2 = 40°. One special case worth remembering: an inscribed angle that subtends a semicircle (a 180° arc) is always exactly 90° — useful for spotting hidden right angles in circle diagrams.
Polygon interior angles — one formula, every polygon
The sum of a polygon’s interior angles depends only on its number of sides, ‘n’: (n − 2) × 180°. Where does that come from? Pick any vertex and draw diagonals to every other non-adjacent vertex — this splits the polygon into (n−2) triangles, and each triangle contributes 180°. A pentagon (n=5) splits into 3 triangles: (5−2)×180 = 540°. A hexagon (n=6) splits into 4: (6−2)×180 = 720°.
Regular polygons — dividing the sum evenly
In a regularpolygon (all sides and angles equal), each interior angle is just the total sum divided by the number of angles: a regular hexagon’s interior angle sum is 720°, divided across 6 equal angles, giving 120° each.
The mistake I see most
Doubling the inscribed angle instead of halving it — getting the central-angle/inscribed-angle relationship backward. The fix: the angle with its vertex at the center is always the bigger one (the full central angle); the angle with its vertex on the circle is always the smaller one (half of it). Identify which vertex type you have before applying the relationship.
Want to work through it together?
If circle vocabulary still feels like a wall of new terms, that clears up fast once you see each term tied to a picture — we’ll build that picture together, one relationship at a time.
