Open an angle at the center and it speaks to an arc and a chord — all three move together.
Stand at the very center of a circle and open an angle. Its two arms are radii, and across from them lies a slice of the rim — an arc. Open the angle wider and that arc grows; the straight chord joining its ends grows too. In one circle these three — central angle, arc, and chord — are locked together: know one and you know them all. This lesson nails down that lockstep, then adds a fourth partner, the distance from the center to the chord. Together they form the chain that powers nearly every circle proof to come.
A central angle is an angle whose vertex is the center O and whose two sides are radii. We write it ∠AOB, where A and B are the points where the radii meet the circle.
The piece of the rim that lies inside the angle — the arc stretching from A around to B on the near side — is the arc the angle subtends. (To subtend just means "to stand under": the angle stands under, or opens onto, that arc.) Pull the arms apart and you sweep out a wider angle and a longer arc together; squeeze them shut and both shrink. The central angle and its arc are two views of the same opening.
A central angle sits at the center and opens onto exactly one arc. Wider angle ⇒ longer arc — always, in the same circle.
Drag the slider to open the central angle. Watch the amber arc and the green chord grow together.
How big is an arc? We measure it the way we measure a turn. Carve the whole rim into 360 equal degrees, one for each degree of a full revolution. Then the measure of an arc is simply the measure of its central angle. A 90° central angle subtends a 90° arc; a 150° central angle subtends a 150° arc. Arc and angle carry the very same number.
Arcs come in three sizes. A minor arc measures less than 180° (the smaller piece). A semicircle is exactly 180° — the arc cut off by a diameter. A major arc measures more than 180° (the larger piece). The minor arc and the major arc on the same two points together make the whole rim, so they always add to 360°:
minor arc + major arc = 360°.
So if a minor arc measures 110°, its major arc measures 360° − 110° = 250°.
An arc's measure (in degrees) is not its length (in units). Two arcs can both measure 60° yet have different lengths — a 60° arc on a big circle is longer than a 60° arc on a small one. Measure tells you the fraction of the circle; length waits for lesson 18.6.
Here is the heart of the lesson. In the same circle (or in congruent circles), these three statements are completely interchangeable:
equal central angles ⇔ equal arcs ⇔ equal chords.
Any one of them forces the other two. Why? Two of the links are easy. Equal arcs are equal central angles — that is just the definition of arc measure from 18.2.2. The interesting link is the chord.
Suppose two central angles are equal: ∠AOB = ∠COD. Look at triangles △OAB and △OCD. Each has two sides that are radii, so OA = OB = OC = OD = r, and their included angles are equal. By SAS (two sides and the included angle), △OAB ≅ △OCD. Congruent triangles have equal third sides, so the chords match: AB = CD. Run the reasoning backward — equal chords give congruent triangles by SSS, hence equal angles — and the chain closes into a perfect loop.
In ⊙O, two central angles are each 70°. Then their arcs each measure 70°, and their chords are equal — by SAS, the two radius-radius-angle triangles are congruent. One equality settled all three.
Flip between two equal central angles and two unequal ones. Read what happens to the arcs and chords.
There is a fourth partner in this dance. The chord-center distance is the perpendicular distance OM from the center O straight down to a chord, meeting it at its midpoint M. From lesson 18.1, dropping that perpendicular makes a right triangle with hypotenuse r, so
half-chord = √(r² − OM²) ⇒ the longer the chord, the smaller OM.
Picture it: a long chord stretches across near the middle of the circle, so it sits close to the center (small OM). A short chord huddles out near the rim, far from the center (large OM). The longest chord of all, the diameter, passes right through O, so its chord-center distance is 0. This extends our chain by one more link:
bigger central angle ⇒ longer arc ⇒ longer chord ⇒ smaller OM,
and in particular equal chords ⇔ equal chord-center distances (chords the same length sit the same distance from the center).
Take r = 6. A 90° central angle gives a chord of length 2·6·sin45° = 6√2 ≈ 8.49, sitting OM = √(36 − 18) = √18 ≈ 4.24 from O. Widen the angle to 120°: the chord grows to 2·6·sin60° = 6√3 ≈ 10.39, while OM shrinks to 6·cos60° = 3. Longer chord, closer to the center — exactly as promised.
Open the angle. As the chord lengthens, watch its distance OM to the center shrink — until at 180° the chord is a diameter and OM hits 0.
One angle at the center sets everything in motion:
bigger central angle ⇒ longer arc ⇒ longer chord ⇒ smaller chord-center distance. Read it in either direction.
A central angle measures 72°. What is the measure of the arc it subtends?
72°. An arc's measure equals its central angle.
A minor arc measures 110°. Find the measure of the major arc on the same two points.
minor + major = 360°, so major = 360° − 110° = 250°.
In a circle, two chords are equal in length. What can you say about their arcs and their central angles?
In the same circle, equal chords ⇒ equal arcs and equal central angles. Equal chords make congruent radius-radius-chord triangles (SSS), so the included central angles are equal, and equal central angles mean equal arcs.
A central angle of 90° sits in a circle of radius 6. Find the length of the chord it cuts.
chord = 2r·sin(½·90°) = 2·6·sin45° = 12·(√2⁄2) = 6√2 ≈ 8.49.
Two chords of one circle are exactly the same distance from the center. Must they be equal in length? Explain.
Yes. half-chord = √(r² − OM²); since r and OM are the same for both, the half-chords match, so the chords are equal. Equal chord-center distances ⇔ equal chords.
The central angle of a semicircle is ____ , and the chord across a semicircle is the ____ .
180°, and the chord is the diameter (length 2r) — the longest chord, sitting OM = 0 from the center.
Six questions to lock it in. Tap the answer you think is right.
This lesson turns one moving part — the central angle — into a whole machine. Open the angle and an arc, a chord, and a chord-center distance all respond in lockstep. The single proof to internalize is the SAS step: two radii and the angle between them pin down the chord, so equal central angles ⇔ equal arcs ⇔ equal chords in any one circle. Everything else here is reading that chain forwards or backwards.
The misconception to head off is the confusion between an arc's measure and its length. A 60° arc is 60° in a thimble and 60° in a stadium, but the two arcs are wildly different lengths. Keep saying "measure is the fraction of the turn; length is units, and it scales with the radius (lesson 18.6)." The second trap is intuition about distance: students often guess a longer chord lies farther from the center. Send them back to half-chord = √(r² − OM²): as the chord grows, OM must shrink, and the diameter (the longest chord) runs right through the center with OM = 0.
Common Core: this lesson supports G-C.A.2 (identify and describe relationships among central angles, arcs, and chords, including the chord-center distance) and G-C.B.5 (the measure of an arc as a fraction of the full circle). The reasoning rests on triangle congruence (SAS / SSS) from Stage 15 and the perpendicular-from-the-center fact from lesson 18.1.