Chapter 12.1 — Introduction to Circles

Three Regions, One Boundary

Any circle splits the entire flat plane it sits on into exactly three regions: the interior (inside the circle), the circumference (the circle itself, its actual boundary), and the exterior (everything outside). A point can belong to only one of these three at a time — there's no overlap and nothing left uncategorized. Deciding which region a given point falls into is purely and simply a matter of comparing its distance from the centre against the radius: closer than the radius means interior, exactly equal to the radius means it sits directly on the circumference, and farther than the radius means exterior — three possibilities in total, covering every single point in the entire flat plane without a single exception anywhere. This single distance comparison is really the entire definition of a circle doing its work again — the same "fixed distance from a fixed point" idea that defines the circle itself is also exactly what sorts every other point in the plane into one of these three regions.

Every Named Part, on One Figure

Every one of these terms describes a piece carved out of the same single circle, just by different cutting tools: a chord alone creates an arc pair and a segment pair; a diameter is simply the special chord that also happens to pass through the centre, splitting the circle into two equal semicircles rather than an unequal major/minor pair; and two radii together, rather than a chord, are what create a sector instead of a segment. Reading the labels "minor" and "major" consistently is worth pausing on, since the same two words apply to arcs, segments, and sectors alike but always mean the same thing each time: minor is always the smaller of the two pieces a chord (for arcs and segments) or a pair of radii (for sectors) splits the circle into, and major is simply whatever larger piece remains once the minor piece is set aside. A circle with only a single chord drawn across it, no diameter, therefore already contains a minor arc, a major arc, a minor segment, and a major segment all at once — four distinct named regions from one single chord.

Segments vs. Sectors — Easy to Mix Up

A segment is bounded by a chord and an arc; a sector is bounded by two radii and an arc. The distinction matters because they're genuinely different regions, not two names for the same thing: a sector always includes the centre as one of its three boundary-defining points, while a segment's boundary never touches the centre at all unless the chord happens to be a diameter. A useful physical way to picture the difference: a sector looks like a slice cut from a round pizza, its two straight edges meeting exactly at the centre where every slice's point comes together; a segment looks instead like the piece left over after a single straight cut is made across a round cake, with one flat edge (the chord) and one curved edge (the arc), and no requirement whatsoever that the straight cut needs to pass anywhere near the centre point at all. When A and B are the two endpoints of a diameter specifically, the chord AB splits the circle into two equal semicircles — the one case where "segment" and "half the circle" mean exactly the same thing. This is really a direct consequence of the diameter's own definition rather than a separate fact needing its own proof: since a diameter passes through the centre, both of the arcs it creates sweep through exactly half the circle's full boundary, and both of the segments it creates likewise split the circle's full interior area exactly in two — no other chord, since it necessarily misses the centre, can ever produce two genuinely equal pieces this way. This is a useful test to keep in mind while reading any circle figure: the moment a chord is described as passing through the centre, every "minor/major" distinction attached to it quietly disappears, since both resulting arcs and both resulting segments are now provably equal rather than one merely being smaller than the other.

From Vocabulary to Worked Problems

Every term defined here — radius, chord, diameter, arc, segment, sector — gets tested directly in Exercise 12.1, before Exercise 12.2, 12.3, 12.4, and 12.5 build genuine theorems on top of this foundation, one after another — starting with how a chord's length relates to the angle it subtends at the centre. This vocabulary-first approach is deliberate: every theorem across the rest of this chapter is stated using these exact terms without redefining them again, so a proof that reads "the perpendicular from the centre to a chord bisects the chord" only makes sense once "chord," "centre," and "bisects" are already fixed, agreed-upon meanings rather than words a reader is still learning to recognise mid-proof.

Where a Circle's Numbers Come From

Unlike every polygon studied in earlier chapters, a circle's entire shape is controlled completely by exactly one single number — its own radius. Two entirely separate circles sharing the same radius are always identical in every single measurable way, no matter exactly where either one is drawn or however it's rotated, since nothing whatsoever about a circle's own basic definition depends on its position or its orientation on the page at all. This single-number simplicity is part of why concentric circles are worth naming specifically: a family of concentric circles is really just one center point paired with an entire range of possible radius values, each distinct radius producing one more circle nested carefully inside or around all the others, all sharing that identical centre. This is also exactly why the phrase "a circle with centre O" is used so consistently throughout the rest of this chapter's problems — naming the centre point immediately fixes which of potentially many concentric circles a given radius, chord, or diameter actually belongs to, avoiding any ambiguity about which circle in a crowded figure a particular measurement refers to.