Chords and Angles in a Circle

A chord is a line segment whose two endpoints both lie on a circle. When a chord is drawn, it divides the circle into two arcs and also creates angles at various points — at the centre, on the circumference, or between intersecting chords. Understanding how these angles relate to the arcs they intercept is one of the most elegant and practically useful topics in Euclidean geometry. The central angle theorem states that the angle subtended by an arc at the centre of a circle is exactly twice the angle subtended by the same arc at any point on the remaining part of the circumference. This single theorem unlocks a cascade of results: that angles in the same segment are equal, that the angle in a semicircle is always 90°, and that the opposite angles of a cyclic quadrilateral are supplementary. These results were systematically proved by the ancient Greeks and compiled in Euclid's Elements around 300 BCE. Today they form the backbone of circle geometry problems in competitive mathematics and are essential tools for proving more advanced results in trigonometry, coordinate geometry, and even computer graphics.

Key takeaways
  • The Central Angle Theorem
  • Angles in the Same Segment Are Equal
  • Angle in a Semicircle (Thales' Theorem)
  • Cyclic Quadrilateral Theorem
  • Intersecting Chords Angle Theorem
  • Chord Length and the Perpendicular from the Centre

The Central Angle Theorem

Let arc AB be any arc of a circle with centre O, and let P be any point on the major arc (the arc not containing AB). Then the central angle ∠AOB = 2 × ∠APB. The proof uses the fact that triangle OAP and triangle OBP are both isosceles (OA = OB = OP = radius), so the base angles are equal. By the exterior angle theorem applied to each triangle, ∠AOB = 2∠APB.

Angles in the Same Segment Are Equal

If two points P and Q both lie on the major arc of chord AB, then ∠APB = ∠AQB. This follows directly from the central angle theorem: both angles equal half the central angle ∠AOB, so they must equal each other. Equivalently, all inscribed angles that intercept the same arc are congruent.

Angle in a Semicircle (Thales' Theorem)

If AB is a diameter of a circle and P is any other point on the circle, then ∠APB = 90°. This is a special case of the central angle theorem: the central angle for a diameter is 180° (a straight angle), so the inscribed angle is 180° ÷ 2 = 90°. This result is attributed to Thales of Miletus and is one of the oldest known geometric theorems.

Cyclic Quadrilateral Theorem

A quadrilateral ABCD is called cyclic if all four vertices lie on a single circle. For any cyclic quadrilateral, opposite angles are supplementary: ∠A + ∠C = 180° and ∠B + ∠D = 180°. Proof: ∠A subtends arc BCD and ∠C subtends arc BAD. Together these arcs make the full circle (360°), so 2∠A + 2∠C = 360°, giving ∠A + ∠C = 180°.

Intersecting Chords Angle Theorem

When two chords PQ and RS intersect inside a circle at point T, the angle at their intersection equals half the sum of the two intercepted arcs: ∠PTR = ½(arc PR + arc QS). This is proved by drawing chord PS; then ∠PTR is an exterior angle of triangle PTS, so ∠PTR = ∠TPS + ∠TSP = ½ arc RS + ½ arc PQ = ½(arc PR + arc QS).

Chord Length and the Perpendicular from the Centre

The perpendicular from the centre O to a chord AB bisects the chord. Conversely, the line from the centre to the midpoint of a chord is perpendicular to the chord. If the radius is r and the perpendicular distance from the centre to the chord is d, then the half-length of the chord is √(r² − d²), giving full chord length = 2√(r² − d²).

The Key Equation

∠AOB=2∠APB\angle AOB = 2 \angle APB
  • \angle AOB — Central angle (degrees or radians): The angle formed at the centre O of the circle by radii OA and OB.
  • \angle APB — Inscribed angle (degrees or radians): The angle formed at point P on the circle by chords PA and PB, where P lies on the arc not containing AB.

Worked Example

Problem. In a circle with centre O, chord AB subtends a central angle of 140°. Point P lies on the major arc. Find: (a) ∠APB, (b) the angle ∠APB if P were instead on the minor arc.

∠APB=70∘ (major arc);∠APB=110∘ (minor arc)\angle APB = 70^\circ \text{ (major arc)};\quad \angle APB = 110^\circ \text{ (minor arc)}

Answer. ∠APB = 70° (major arc); ∠APB = 110° (minor arc)

Figure
Diagram
Central Angle Theorem & Angles in a CircleOABchord AB140°P70° = ½ × 140°● Blue: central angle ∠AOB = 140°● Green: inscribed angle ∠APB = 70°● Red: chord AB
Origins · 1 minThales of Miletus and how this was discovered
TO
Thales of Miletus
-624–-546 · Greek

Thales is credited with the first recorded deductive proof in mathematics: that the angle inscribed in a semicircle is always a right angle (90°). This result, now called Thales' Theorem, is a special case of the central angle theorem with ∠AOB = 180°. He is also said to have used similar triangles to measure the heights of Egyptian pyramids. Ancient sources claim Thales predicted a solar eclipse in 585 BCE — an astonishing feat for the time — and that he once fell into a well while gazing at the stars, prompting a local woman to remark that he could see what was in the heavens but not what was at his feet.

  • Tangent Lines to Circles — The tangent-chord angle theorem (angle between a tangent and a chord equals the inscribed angle in the alternate segment) extends the chord-angle results studied here.
  • Cyclic Quadrilaterals — Cyclic quadrilaterals rely directly on the fact that opposite inscribed angles subtend arcs that together form the full circle, making their angle sums 180°.
  • Arc Length and Sector Area — Arc measure in degrees or radians is the underlying quantity that both the central angle and inscribed angle reference, connecting angle theorems to metric properties of circles.
  • The Sine Rule — The extended sine rule — a/sin A = 2R — connects side lengths of a triangle to the circumradius R, and is proved using the inscribed angle theorem.