Circle Definitions & Properties

SPEC G9 GEOMETRY & MEASURES F+H

Nine named parts of a circle — plus three preliminary properties that set up Circle Theorems in Lesson 4.10

What you will learn

By the end of this lesson you will:

Identify and name the nine parts of a circle : centre, radius, diameter, chord, circumference, tangent, arc, sector, segment

Distinguish the easily confused pair — sector vs segment — cleanly

State three key properties that underpin every circle theorem: tangent radius; perpendicular from centre bisects chord; two tangents from an external point are equal

Use the correct circle vocabulary in written answers so reasoning marks are not lost

A "vocabulary" lesson, but not a soft one

On a recent AQA paper, Q1 asked students to name a sector and a segment. Over a third of students scored zero — most wrote "sector" for the segment. Learning the words properly is how you don't lose 2 easy marks.

Why this lesson matters

Almost every circle question — whether on length, area, or geometric reasoning — uses one of these nine words. The specification (G9) explicitly lists them as content you must identify and apply. They split into three natural groups:

GroupWordsWhat they describe

The circle itselfCentre, Radius, Diameter, CircumferenceFundamental points and distances on the circle

Lines that meet itChord, TangentStraight lines that cross or touch the circle

Parts & regionsArc, Sector, SegmentPortions of the circumference or the circle's interior

Exam phrasing you will see

"Name the part labelled X." "Complete the sentence: a ___ touches the circle at exactly one point." "State one property of a tangent." Two marks each, ~60 seconds each, rarely less than 2 marks per paper at Higher.

Centre, Radius, Diameter

Three fundamentals

Centre (labelled O): the point equidistant from every point on the circle. Radius: a line from the centre to any point on the circle; its length is normally called r. Diameter: a line joining two points on the circle through the centre; length 2r. Here AB is a diameter.

Plural note

Plural of radius is radii (pronounced "ray-dee-eye"). You will see this word a lot in Circle Theorems.

Chord

A chord is a straight line joining any two points on the circle. It does not have to pass through the centre.

Relationship with diameter

Every diameter is a chord (the special chord that passes through the centre). But not every chord is a diameter. The diameter is the longest possible chord.

Circumference

The circumference is the total perimeter of the circle — the curved boundary itself. Its length is given by the formula C = 2πr or equivalently C = πd.

Word use

Sometimes "circumference" refers to the curve (as in "a point on the circumference"); sometimes to the length of the curve (as in "calculate the circumference"). Both uses are standard and accepted.

Tangent

A tangent is a straight line that touches the circle at exactly one point and does not cross it. That single shared point has its own name: the point of contact. On the diagram, T is the point of contact between the tangent and the circle.

Preview — a key property

The tangent is always perpendicular to the radius drawn to the point of contact. We'll use this three slides from now. Right-angle at the point of contact = tangent's signature move.

Arc

An arc is a section of the circumference between two points. Any two points on the circle cut it into a minor arc (the shorter path) and a major arc (the longer path).

Mark-scheme language

Whenever there is any ambiguity about which arc is meant, always specify "minor arc AB" or "major arc AB". Don't rely on "arc AB" by itself — examiners expect explicit wording and a vague answer can lose the reasoning mark.

Sector

A sector is a region of the circle bounded by two radii and the arc between their endpoints. Shape: a "pie slice".

Two straight edges

Both straight edges of a sector are radii, so they both start at the centre. That is the signature feature: if one edge doesn't reach the centre, it's not a sector.

Segment

A segment is a region of the circle bounded by a chord and an arc. Shape: a "knife cut" off the circle.

One straight edge — not a radius

A segment has only one straight edge (the chord), and that edge does not touch the centre. Just like arcs, a chord cuts the circle into a minor segment (smaller) and a major segment (larger).

Critical: Sector vs Segment

This is the single biggest cause of lost marks on circle-vocabulary questions. Lock in the difference with the side-by-side below.

Two radii + an arc. Think: "pie slice". Both straight edges go to the centre.

One chord + an arc. Think: "knife cut". The straight edge does NOT touch the centre.

⚠️ Recent AQA evidence. In a recent AQA paper, Q1 asked students to name a sector (Region B) and a segment (Region A). More students wrote "sector" for the segment than wrote "segment". Over a third scored 0/2. Memorise the pair using a single trigger: does the straight edge go to the centre? — yes → sector; no → segment.

Worked Example 1

01Name the shaded region 2 marks

The diagram shows a circle, centre O, and three straight lines. Use one word to describe each shaded region. Choose from: arc, chord, sector, segment, tangent.

M

For each region, ask: "Is the straight edge a radius (going to the centre) or a chord (not touching the centre)?" One radius-bounded edge → sector; chord-bounded edge → segment.

C

Region A has a single chord as its straight edge — the chord does not pass through O. So A is a segment. Region B has two straight edges, both going to the centre (they are radii). So B is a sector.

E

Region A = segment. Region B = sector. Exam tip: name the part in full words as they appear in the word bank. Spellings matter — "segmant" or "secter" can lose the mark.

Property 1: Tangent Radius

The property

At the point of contact T, the tangent is perpendicular to the radius. Whenever a tangent appears in a question, draw the radius to the point of contact. The 90° angle you create is almost always how the problem is solved.

Mark-scheme phrasing

"Tangent perpendicular to radius at point of contact." or "Angle between tangent and radius = 90°." Either is accepted.

Property 2: Perpendicular from centre bisects chord

The property

If you drop a perpendicular from the centre to a chord, it hits the chord at its midpoint M. That is, PM = MQ. Equivalently: the line from the centre that bisects a chord is perpendicular to it.

Why it's useful

This property turns chord problems into right-angled triangles. Triangle OPM (or OQM) is right-angled at M, with OP as the hypotenuse. Since OP is a radius (length r), Pythagoras gives r 2 = OM 2 + PM 2. If you know the radius and either OM or PM, you can find the other — a very common exam move.

Property 3: Two tangents from an external point are equal

The property

From a point P outside the circle, there are exactly two tangents to the circle. The two tangent lengths PT 1 and PT 2 are equal.

Extra consequences

Triangles OPT1 and OPT2 are congruent (RHS: both right-angled at T; common hypotenuse OP; equal sides OT1 = OT2 = radius). So the line OP bisects both angle T1PT2 and angle T1OT2.

Worked Example 2

02Complete the sentences 3 marks

Choose a word from the list below to complete each sentence. Each word may be used at most once.

Word bank

arc centre chord circumference diameter radius sector segment tangent

(a) The length of the ________ is double the length of the radius.

(b) A ________ is a region bounded by a chord and an arc.

(c) A radius meets a ________ at a right angle.

M

For each sentence, match the description to one of the nine defined parts. Read the sentence for the signature feature of each part.

C

(a) "Double the radius" describes the diameter (d = 2r).

(b) "Bounded by a chord and an arc" is the exact definition of a segment.

(c) "Radius meets at a right angle" is Property 1 — the tangent.

E

(a) diameter (b) segment (c) tangent. One mark per correct word. Write the word in full — abbreviations like "dia" or "tang" will not score.

Common Errors

Students often writeCorrect versionReason

"Sector" for a region bounded by a chordSegmentSector has two radii; segment has a chord

"Chord" for the curved part between two pointsArcChord is straight; arc is curved

"Diameter" for any chordChord (unless it passes through O)Diameter is the special chord through the centre

"Radius" for the diameter's lengthDiameter = 2 × radiusKnow both lengths: d = 2r, so r = d/2

"Tangent" for a chordTangent touches at 1 point; chord meets at 2 pointsCounting points of intersection

Tangent makes acute/obtuse angle with radiusTangent radius at point of contact (90°)Property 1 is non-negotiable

⚠️ Exam style: if the question gives a word bank, always pick your answer from the list. Synonyms and near-spellings (e.g. "slice" for sector, "secter" for sector) do not score.

Summary

The nine parts of a circle

PartIn one line

CentreThe point equidistant from all points on the circle

RadiusA line from centre to the edge; length r

DiameterA chord through the centre; length 2 r

ChordA straight line joining two points on the circle

CircumferenceThe perimeter of the circle; C = 2π r

TangentA line that touches the circle at exactly one point

ArcA portion of the circumference

SectorRegion bounded by two radii and an arc (pie slice)

SegmentRegion bounded by a chord and an arc (knife cut)

Three properties

Tangent radius at point of contact

Perpendicular from centre bisects chord

Two tangents from an external point are equal

One-question trigger

Sector vs segment: "Does the straight edge touch the centre?" Yes → sector. No → segment.

⚠️ Next lesson: Circle Theorems (4.10) builds directly on Properties 1 and 2. Mastering this vocabulary is the gateway to the 4–5 mark theorem questions at Higher.