Is this the angle at the centre?

What this sequence teaches

The angle at the centre of a circle is twice the angle at the circumference, when both angles are drawn standing on the same chord. Three critical features define when a marked centre-and-circumference pair satisfies the theorem:

• CF1 (vertex configuration): exactly one angle has its vertex at the centre of the circle; the other has its vertex on the circumference.
• CF2 (same chord): both angles are formed from the same chord — their boundary rays end at the same two points on the circle.
• CF3 (matching branch): if the centre angle is non-reflex, the circumference vertex sits on the major arc. If the centre angle is reflex, the circumference vertex sits on the minor arc.

The sequence is six Examples (NPPNPN). The opening N attacks CF1; the closing N attacks CF2; a middle pair (Example 4 → Example 5) demonstrates CF3 by pivoting from a branch-mismatch negative to a reflex-branch positive. This deviates from the default NPPPNN shape for three-CF atoms. The deviation is deliberate — the reflex branch is non-intuitive enough that it must appear as a positive in the teaching sequence rather than being deferred to testing.

Colour convention. In every diagram, the teal wedge marks the candidate centre angle — the angle being claimed as “at the centre”. The purple wedge marks the candidate circumference angle. The colours describe what each angle is claimed to be, not what it actually is. The student’s job is to verify whether the teal vertex really does sit at the centre marker, whether both angles use the same chord, and whether the branches match. A small “+” cross marks the geometric centre on every diagram for reference.

Frame by frame

Example 1 (N) — same segment. Two angles, both vertices on the circumference, both formed from the same chord at the bottom of the circle. The teal wedge marks the candidate centre angle — but its vertex is on the circumference, not on the centre marker. A different circle theorem (angles in the same segment) makes both inscribed angles equal — but neither is at the centre. Fails CF1.

1 → 2 animation. The candidate centre vertex (teal) translates straight down from the circumference to the actual centre point. As it moves, the angle it subtends grows from 40° to 80° (the chord is seen from closer in). The left circumference vertex stays anchored throughout.

Example 2 (P) — chords-crossing, acute centre. CF1 now satisfied: one centre, one circumference. Centre angle 80°, inscribed angle 40°. The chord from the left circumference vertex to the right-hand chord endpoint passes through the centre wedge. This “chords-crossing” flavour is one of the two visual archetypes the theorem appears in.

Example 3 (P) — Star Trek, obtuse centre. Rotated orientation, larger centre angle (140°), inscribed angle on the major arc directly opposite the chord. The classic textbook kite-shape arrangement. The variation between Example 2 and Example 3 is deliberately maximal — different orientation, different magnitude, different visual flavour.

3 → 4 animation. At the centre vertex, the shading swaps from the non-reflex angle to the reflex angle on the other side of the chord. The boundary rays are unchanged.

Example 4 (N) — reflex shading, P on major arc. Same boundary rays as Example 3, but the centre wedge is now the reflex 220°. The circumference vertex is still on the major arc. A reflex centre angle must pair with a circumference vertex on the minor arc. Fails CF3.

4 → 5 animation. The circumference vertex slides along the circle from the major arc to the minor arc, passing through one of the chord endpoints. The wedge magnitude changes 70° → 110° because the inscribed angle on the minor arc subtends the major arc, which corresponds to the reflex centre.

Example 5 (P) — reflex centre, P on minor arc. Reflex 220° at the centre paired with an inscribed 110° on the minor arc. The alternative branch of the theorem. Students often dismiss this as wrong because it doesn’t look like the textbook example — but it satisfies all three CFs.

5 → 6 animation (multi-beat). This transition runs in stages with a pause between them. Beat 1: the centre shading flips reflex back to non-reflex, then the circumference vertex slides along the circle through one chord endpoint to the bottom of the circle. The Example briefly returns to a canonical Star Trek configuration — a transient internal positive state. The pause at this configuration is deliberate. Beat 2: one of the boundary rays of the circumference angle sweeps along the circle, dragging its endpoint away from the centre’s matching endpoint. The two angles now use different chords.

Example 6 (N) — different chord. Star Trek configuration at first glance, but the right-hand boundary endpoint at the circumference is in a different place on the circle from the centre’s right-hand endpoint. The two angles are not formed from the same chord. Fails CF2.

What’s not covered here

The gap form of the chord-mismatch misconception (P’s endpoint slightly off the centre’s endpoint, rather than dramatically) is deferred to the testing sequence. Once students have seen the dramatic version in Example 6, the sophisticated version becomes a useful test of how precise their understanding of “same chord” is.

The “almost at the centre” misconception (a vertex inside the circle but not at the geometric centre) is deferred to testing. It’s intuitive to spot once students are looking for it, but easy to miss when surface features are otherwise canonical.

The sophisticated branch-mismatch (non-reflex centre with P on the minor arc) is deferred to testing. The teaching sequence shows the cruder form (reflex centre with P on the major arc) in Example 4 because it pivots cleanly into the reflex-branch positive in Example 5.

Running the sequence

Click forward to step through the six Examples. Animations between adjacent frames are reversible. The verdict on each Example is hidden until reveal — ask students to commit to a verdict and a reason before clicking. The 4 → 5 transition is the highest-leverage moment in the sequence; it’s the visual proof that the reflex branch is a legitimate positive, not a contrived edge case. Don’t rush past it. The 5 → 6 multi-beat transition deserves a similar pause — Example 6 looks like a positive, and the only way to call it correctly is to check chord endpoints one by one.

About the testing sequence

Ten items, presented in randomised order on each load. Six positives, four negatives. The positives deliberately vary across the dimensions students will encounter: orientation of the figure, magnitude of the centre angle (from very acute 40° through reflex 240°), and the two visual archetypes (chord-crossing and Star Trek). The negatives target one critical feature each in either crude or sophisticated form. Students should leave with the sense that surface variation — orientation, angle size, where the chord sits — doesn’t matter; only the three critical features do.

What each item is diagnosing

Positives:

• Standard Star Trek, acute centre 60°. The textbook configuration. A student who can’t call this positive isn’t ready for the rest of the pool.
• Star Trek, obtuse centre 130°. Rotated, larger magnitude. Probes whether students rely on canonical orientation.
• Chords-crossing, centre 100°. The other visual archetype. Students sometimes call this a no because “the chord crosses the wedge” feels wrong.
• Reflex centre 240°, P on minor arc. The alternative branch of the theorem.
• Very acute centre 40°. Pushes magnitude to one extreme. Probes whether students dismiss narrow angles.
• Very obtuse centre 160°, chord near-diameter. Pushes magnitude to the other extreme.

Negatives:

• Same segment. Crude form of the vertex-configuration misconception — both vertices on the circumference.
• Almost at the centre. Sophisticated form of the vertex-configuration misconception — one vertex is inside the disc, near the centre but not on it.
• Different chord. The centre uses one chord, the circumference vertex uses a different chord. The mismatch is visible but requires tracing the rays.
• Non-reflex centre, P on minor arc. Sophisticated branch-mismatch. Both wedges look “normal sized” and the configuration looks canonical at first glance. The trap is in noticing that the wedge and the circumference vertex are on the same side of the chord rather than opposite sides.

Common confusions

A student who fails the “same segment” item is reaching for a “the two angles look paired so it must be the theorem” rule. Send them to CF1: which of the two angles has its vertex at the centre?

A student who fails the “almost at the centre” item is treating “inside the circle” as equivalent to “at the centre.” Ask: is the vertex at the centre, or just close to it? The geometric centre is a single point.

A student who fails the “different chord” item has CF1 and CF3 but isn’t tracking CF2. Ask: trace each ray. Do both angles end at the same two points on the circle?

A student who fails the sophisticated branch-mismatch is the most common case. Ask: where is the wedge relative to the chord? Where is the vertex relative to the chord? They should be on opposite sides for a non-reflex centre angle.

A student who fails the reflex positive is dismissing the alternative branch. Send them to Example 5 of the teaching sequence and ask: what is the centre angle here? What’s the circumference angle? Does 2 × 110° give 220°?

Discussion prompts

1. Two of the negatives have vertex-configuration as their problem. In one, the misconception is dramatic — both vertices on the circumference. In the other, the misconception is subtle — one vertex is inside the disc, near the centre but not at it. Why is the subtle version harder to call?
2. Two of the items show a Star Trek configuration with a non-reflex centre angle. One is a yes; one is a no. What’s the geometric difference, and why does that difference matter for the theorem?
3. The pool includes a positive where the centre angle is reflex and the circumference vertex sits on the minor arc. Many students will call this a no. What about the theorem’s statement, as usually written in a textbook, makes this configuration look wrong even though it’s right?

Reading the summary

Each item is annotated with two channels of information: a green or red tint indicates whether the student answered correctly; a tick or cross in the corner indicates whether the marked configuration is in fact the angle-at-centre relationship. The two channels can disagree — that disagreement is the diagnostic information worth probing. A student who calls every Star Trek positive correctly but misses the reflex positive is showing branch-blindness; a student who calls the reflex positive correctly but misses the sophisticated branch-mismatch negative is over-generalising in the opposite direction.