Circles & their properties
The idea
A circle is the set of all points at a fixed distance — the radius — from a center, and that single constraint generates a surprising web of theorems about chords, tangents, arcs, and angles. The headline result is the inscribed angle theorem: an angle whose vertex sits on the circle measures exactly half the central angle intercepting the same arc. Arc length and sector area follow from proportional reasoning you already own — an arc's fraction of the whole circle equals its central angle's fraction of 360°.
Approach circle problems by asking where each angle's vertex sits: at the center (the angle equals its arc), on the circle (half its arc), or at a point of tangency (a radius meets a tangent at 90°). For lengths and areas, form the fraction angle/360° and multiply by the circumference 2πr or the area πr². The common slip is treating an inscribed angle as equal to its central angle; the inscribed vertex sits farther from the arc, and its angle is exactly half — forgetting the halving doubles every downstream answer.
Worked example
Points A, B, and C lie on a circle with center O and radius 9 cm, and the central angle AOB measures 100°. Find the inscribed angle ACB, the length of minor arc AB, and the area of sector AOB, giving lengths and areas exactly and to one decimal place.
- Angle ACB is inscribed and intercepts the same arc AB as the central angle, so it measures half of 100°, which is 50°.
- Arc AB takes up 100/360 = 5/18 of the circle. The full circumference is 2π × 9 = 18π cm, so the arc length is (5/18) × 18π = 5π ≈ 15.7 cm.
- The sector uses the same fraction of the full area π × 9² = 81π cm²: area = (5/18) × 81π = 22.5π ≈ 70.7 cm².
- Sanity-check with a quarter circle: 90° would give an arc of 4.5π ≈ 14.1 cm and a sector of 20.25π ≈ 63.6 cm², and both answers for 100° sit slightly above those values, as they should.
Answer. Angle ACB = 50°, arc AB = 5π ≈ 15.7 cm, and sector AOB has area 22.5π ≈ 70.7 cm².
Check your understanding
- Why is an inscribed angle exactly half its central angle, and what special case appears when the inscribed angle intercepts a semicircle?
- How does the single fraction angle/360° unify arc length and sector area into one idea?
- Why must a tangent line be perpendicular to the radius drawn to the point of tangency?
- How would the arc length and sector area change if the radius doubled while the central angle stayed at 100°?
Build the foundations first
Circles & their properties builds on these concepts. If any feel shaky, start there.