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Home›Geometry and drawings›Circles

Parts of a circle: radius, diameter, chord, arc, sector and tangent

Draw any two chords on a round plate. The lines that cut each chord in half at right angles cross at exactly one point, and that point is the centre.

Where is the middle of a round plate?

Suppose you cut a paper circle out of a magazine and want to push a pin through its exact middle. A circle has no corners to measure from, so a ruler on its own is little help. Folding is. Fold the paper exactly in half so that the two halves cover each other, and the crease runs straight through the centre. Unfold it, fold it in half a different way, and the second crease crosses the first at the centre. Euclid’s Elements does the same job with straight lines instead of folds. It is the very first proposition of Book III, the book about circles: to find the centre of a given circle.

One distance from one point

A circle is the set of all points in a plane that lie at the same distance from one fixed point. That fixed point is the centre, usually called O. A segment from the centre to any point of the circle is a radius, and its length is written r. Since every point of the circle is the same distance away, all the radii of one circle are equal, and there are infinitely many of them: one for each point. A segment that joins two points of the circle and passes through the centre is a diameter, d. It is two radii laid end to end, so d = 2r and r = d ÷ 2. A circle with radius 7 cm has diameter 14 cm. A compass draws exactly this definition. Its point stays in the centre while the pencil stays one radius away.

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Drag to turn. Scroll or pinch to zoom.

This circle has its centre at the origin and radius 1. Drag the point anywhere on the circle. Its coordinates change all the time, but the bottom line of the panel, cos² + sin² (which is x² + y² for this point), stays at 1. That equation is the distance rule in another form: the point never leaves distance 1 from the centre.

Chords, arcs, sectors and segments

A chord is any segment whose two ends lie on the circle. A diameter is a chord too: the one that passes through the centre. It is also the longest chord; Euclid proves in Book III, Proposition 15 that the diameter is the greatest straight line in a circle. Two points on a circle also split the curve itself into two pieces. Each piece is an arc. The shorter one is the minor arc and the longer one the major arc. When the two points are the ends of a diameter, both arcs are semicircles. An arc from A to B is written AB with a small curved mark over the letters.

Two kinds of region come from these pieces. A sector is the region cut off by two radii and the arc between them, the shape of a slice of pizza. The angle between the two radii is the central angle of the sector. A segment of a circle is the region between a chord and its arc. Cut a round loaf of bread with one straight cut, and the piece that falls away is a segment. If the cut is a diameter, the segment is a half circle, and a half circle is also a sector with a central angle of 180°.

Lines that meet the circle in one point or two

A straight line can meet a circle in two points, in one point, or not at all. A line through two points of the circle is a secant, and the part of it inside the circle is a chord. A line that meets the circle at exactly one point is a tangent, and the meeting point is the point of tangency. Euclid’s definition says that a line touches a circle when it meets the circle and, extended either way, does not cut it. A bicycle wheel standing on a flat road shows the idea well: the road meets the tyre at one point and never goes inside it. Unit 6 proves that the radius drawn to that point is always at right angles to the tangent.

Left: the centre O, a blue radius r, the diameter AB (d = 2r) and a green chord CD. Right: an orange sector, a blue arc, a green segment between a chord and its arc, and a purple tangent touching the circle at T, at right angles to the dashed radius.

Finding the centre with two chords

Back to the paper circle, this time without folding. Draw any chord AB and mark its midpoint M. Through M draw the line at right angles to AB, the perpendicular bisector of the chord. Every point on that line is the same distance from A as from B. The centre is such a point, because OA and OB are both radii, so the centre lies somewhere on this line. Now repeat with a second chord CD that is not parallel to the first. The centre lies on both lines, and two lines that are not parallel cross only once. Their crossing point is the centre. Euclid states the key fact as a corollary to Book III, Proposition 1: if a straight line cuts a chord into two equal parts at right angles, the centre of the circle lies on that line.

Two chords, AB and CD, each with its perpendicular bisector drawn dashed. Matching tick marks show the equal halves. The two bisectors cross at the centre O.
The same right angle measures chords. In a circle of radius 5 cm, take a chord whose midpoint M is 3 cm from the centre. Triangle OMA has a right angle at M, OA = 5 and OM = 3, so AM = 4 by the Pythagorean theorem, and the whole chord is 8 cm. The further a chord is from the centre, the shorter it gets; at distance 0 it is the diameter.
BackgroundEuclid’s own definition, Book I Definition 15 in the D. E. Joyce edition, calls a circle a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure equal one another. Joyce’s notes point out two differences from today. For Euclid the circle is the flat region; the curved line around it is the circumference, while modern books usually mean the curve itself when they say circle. And the English word centre comes from a Greek word that also means a prod or a poker, a reference to the fixed, pointed leg of the compass. Euclid had no separate word for radius at all. He wrote “the line from the centre”, and radius is the translator’s shorter word.
CentreThe fixed point; every point of the circle is the same distance from it
Radius rSegment from the centre to the circle
Diameter dChord through the centre; d = 2r, the longest chord
ChordSegment joining two points of the circle
ArcPart of the circle between two of its points
SectorRegion between two radii and their arc
SegmentRegion between a chord and its arc
TangentLine that meets the circle at exactly one point
SecantLine that meets the circle at two points
  1. A circle has radius 7 cm. How long is its diameter?
  2. A circle has diameter 18 cm. How long is its radius?
  3. Which is the longest chord of a circle?
    • a diameter
    • a radius
    • a chord near the edge
    • all chords are the same length
  4. How many radii does a circle have?
    • infinitely many
    • 2
    • 4
    • 360
  5. The region between a chord and its arc is called a:
    • segment
    • sector
    • semicircle only
    • secant
  6. A line that meets a circle at exactly one point is a:
    • tangent
    • secant
    • chord
    • diameter
  7. A circle has radius 5 cm. A chord’s midpoint is 3 cm from the centre. How long is the chord?
  8. A circle has radius 13 cm. A chord is 5 cm from the centre. How long is the chord?
  9. A chord 16 cm long sits in a circle of radius 10 cm. How far is it from the centre?
  10. Is every diameter a chord?
    • yes, it is the chord through the centre
    • no, chords never pass through the centre
    • only in large circles
    • only when it is drawn across
  11. The perpendicular bisector of any chord passes through:
    • the centre of the circle
    • one end of the chord
    • the point where a tangent touches
    • no special point
Common mistakes
  • “A diameter is not a chord, they are different things.” A diameter is the special chord that passes through the centre. It is a chord and also the longest one.
  • “Arc AB and chord AB are the same length, since both go from A to B.” The chord is the straight path and the arc is the curved one. A straight segment is the shortest path between two points, so the arc is always longer.
  • “A sector and a segment are two names for one shape.” A sector has a corner at the centre and two straight sides; a segment has one straight side, a chord, and no corner at the centre. Only the half circle is both.
Teacher's noteIn the 2022 Korean national curriculum the centre, radius and diameter come in primary school, grades 3 to 4, together with drawing circles of different sizes with a compass; the notes ask teachers to link the compass method with the properties of the circle. Chord, arc, sector, segment, secant, tangent and point of tangency are on the term list for middle school, grades 1 to 3. The pair that causes most trouble is arc and chord, because both run from A to B. Have learners lay a thread along an arc and a ruler along the chord, then compare the two lengths.
Key points
  • A circle is all points at distance r from the centre O. Every radius has length r, and d = 2r.
  • A chord joins two points of the circle; the diameter is the longest chord.
  • Arc: a piece of the circle. Sector: two radii and an arc. Segment: a chord and an arc.
  • A secant meets the circle twice, a tangent once. The perpendicular bisector of any chord passes through the centre.
Sources for this course (5)
  • Curriculum: 교육부 고시 제2022-33호 [별책 8] 수학과 교육과정 (ncic.re.kr/inv/org/download.do?year=2022&seq=10003559&orgType=ogi4, PDF read 2026-10-19: 초등학교 3~4학년군 [4수03-06] 원의 중심, 반지름, 지름을 이해하고 그 성질, [4수03-07] 컴퍼스로 여러 가지 크기의 원 그리기(해설: 원의 성질과 연결); 용어 원의 중심, 반지름, 지름; 초등학교 5~6학년군 [6수03-15] 원 모양 물체의 원주와 지름을 측정해 원주율이 일정함을 알고 근삿값 사용(해설: (원주) ÷ (지름), 근삿값 3.14), [6수03-16] 원주와 원의 넓이(해설: 지름과 원주율로 원주를 측정 없이 구함, 직사각형의 넓이를 구하는 방법을 이용해 원의 넓이), 고려 사항: 원주율에 대한 양감, 복잡한 계산은 계산기; 용어 원주, 원주율; 중학교 1~3학년군 [9수03-06] 부채꼴의 중심각과 호의 관계, 호의 길이와 넓이, [9수03-18] 원의 현에 관한 성질과 접선에 관한 성질을 이해하고 정당화, [9수03-19] 원주각의 성질을 이해하고 정당화; 용어 부채꼴, 중심각, 호, 현, 활꼴, 할선, 접선, 접점, 접한다, 원주각; 고려 사항: 복잡하게 변형된 평면도형의 넓이와 둘레는 다루지 않음, 원과 비례에 관한 성질은 다루지 않음, '접선의 길이'는 교수·학습 상황에서 사용 가능)
  • Textbook: OpenStax, Contemporary Mathematics, 10.4 Polygons, Perimeter, and Circumference (C = πd = 2πr, π = 3.141592654… is the ratio of circumference to diameter, d = 10 cm gives about 31.42 cm; Archimedes 287 to 212 BCE; inscribed and circumscribed perimeters for a circle of diameter 1: 4 sides 2.8284 and 4.00, 8 sides 3.0615 and 3.3137, 16 sides 3.1214 and 3.1826, 32 sides 3.1365 and 3.1517, 64 sides 3.1403 and 3.1441, 96 sides: openstax.org/books/contemporary-mathematics/pages/10-4-polygons-perimeter-and-circumference)
  • OpenStax Prealgebra 2e, 9.5 Solve Geometry Applications: Circles and Irregular Figures (d = 2r, C = 2πr = πd, A = πr², 3.14 or 22/7 for π and the calculator π key gives slightly different answers, a circumference of 47.1 cm gives a diameter of about 15 cm, a track shaped like a 105 m by 68 m rectangle with a semicircle at each end: openstax.org/books/prealgebra-2e/pages/9-5-solve-geometry-applications-circles-and-irregular-figures)
  • Euclid’s Elements, D. E. Joyce edition, Clark University (Book I Defs. 15 to 18 and guide: circle as a plane figure, the word centre from a Greek word for a prod or poker, Euclid has no word for radius; Book III Def. 2 touch, Def. 6 segment, Def. 9 stand upon, Def. 10 sector; Prop. 1 and corollary: the centre lies on the line that cuts a chord into equal parts at right angles; Prop. 3; Prop. 15 the diameter is greatest; Prop. 16 corollary; Prop. 17 tangent from a point; Prop. 18 radius to the point of contact perpendicular to the tangent; Prop. 20 angle at the centre double the angle at the circumference; Prop. 21 angles in the same segment equal; Prop. 22 opposite angles of a quadrilateral in a circle; Prop. 28; Prop. 31 angle in a semicircle right, guide: often called Thales’ theorem: mathcs.clarku.edu/~djoyce/elements/bookIII/bookIII.html)
  • History for story blocks only: MacTutor History of Mathematics, University of St Andrews, Archimedes of Syracuse (287 to 212 BC; Measurement of the Circle: π between 3 10/71 and 3 1/7 from inscribed and circumscribed 96-gons; On the sphere and cylinder: the surface of a sphere is four times a great circle: mathshistory.st-andrews.ac.uk/Biographies/Archimedes/) and A history of Pi (Rhind papyrus about 1650 BC with good evidence for 4 × (8/9)² = 3.16; Archimedes 223/71 < π < 22/7 from 96 sides; Zu Chongzhi (430 to 501) 355/113; William Jones first used π with its present meaning in 1706, Euler adopted it in 1737: mathshistory.st-andrews.ac.uk/HistTopics/Pi_through_the_ages/).

Checked: 2026-10-19