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Home›Geometry and drawings›Circles

Central and inscribed angles

Join any point of a circle to the two ends of a diameter and the angle at that point is exactly 90°. Euclid proved it as Book III, Proposition 31.

Three propositions from Euclid’s book of circles

BackgroundBook III of Euclid’s Elements, the book about circles, contains a short chain of results that school courses still teach. Proposition 20 says that in a circle the angle at the centre is double the angle at the circumference when both stand on the same arc. Proposition 21 uses it at once: angles in the same segment are equal to one another. Proposition 31 says that the angle in a semicircle is a right angle, and the D. E. Joyce edition notes that this part is often called Thales’ theorem. Each proof is only a few lines long and uses two facts from Book I: the base angles of an isosceles triangle are equal (Proposition 5), and an exterior angle of a triangle equals the two opposite interior angles together (Proposition 32).

First the names. Take two points A and B on a circle with centre O. The angle AOB, with its vertex at the centre, is a central angle. Now take a third point C on the circle, on the other side from the arc AB. The angle ACB has its vertex on the circle and both its sides are chords. It is an inscribed angle, and Euclid says it stands upon the arc AB. A central angle and an inscribed angle that stand on the same arc are the pair the theorem talks about.

Why the inscribed angle is half the central angle

Here is Euclid’s proof for the usual picture, where the centre lies inside the angle ACB. Draw the line from C through O, and let it meet the circle again at D. Triangle OCA is isosceles, because OC and OA are both radii, so its angles at C and at A are equal; call each one x. The angle AOD is an exterior angle of that triangle, so it equals x + x = 2x. The same argument on the other side, with triangle OCB, gives angle BOD = 2y, where y is the angle OCB. Add the two halves. The central angle AOB is 2x + 2y, and the inscribed angle ACB is x + y. The central angle is twice the inscribed angle.

Left: central angle AOB (blue) and inscribed angle ACB (orange) on the same arc AB. The dashed line from C through O to D splits the figure into two isosceles triangles, as in Euclid’s proof. Right: AB is a diameter, so the angle at C, and at any other point C′ of the circle, is 90°.

If the centre falls outside the angle ACB, the same two triangles appear, but you subtract instead of adding. Euclid handles that case too, at the end of Proposition 20. Either way the result holds: an inscribed angle is half the central angle on the same arc.

Every inscribed angle on one arc is the same

Slide C anywhere along the same arc. The central angle AOB does not move, so neither does its half. All inscribed angles standing on the same arc are equal, which is Proposition 21. A photographer anywhere on that arc sees the stage AB under the same angle. With two points C and D on opposite arcs of the chord AB, the two inscribed angles add to 180°, because their central angles go all the way round and add to 360°. That is why the opposite angles of a quadrilateral with all four corners on a circle add to 180° (Euclid, Proposition 22).

The angle in a semicircle

Now let AB be a diameter. The central angle AOB is a straight angle, 180°, so every inscribed angle on it is 90°. That is Proposition 31. It also works backwards: if C lies on the circle and angle ACB is 90°, the central angle on AB is 180°, so AB must be a diameter. This gives a way to find the centre of a round board with a set square. Put the right-angled corner on the edge, mark where the two arms cross the edge, and the line between the marks is a diameter. Do it twice, and the two diameters cross at the centre.

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A and B are the ends of a diameter of a circle with radius 1.25 (the circle itself is not drawn), and C starts on that circle, so angle C reads 90°. Drag C a little upwards, outside the circle: angle C drops below 90°. Drag it down towards AB, inside the circle, and it grows past 90°. Then put C at the top of the circle, straight above the middle of AB: 90° again, with 45° at A and at B. Euclid’s Proposition 31 says the same: the angle in a segment greater than a semicircle is less than a right angle, and in a smaller segment it is greater.
A right triangle always fits in a circle whose diameter is its hypotenuse. A triangle with legs 6 and 8 has hypotenuse 10, so the circle through its three corners has radius 5, centred at the midpoint of the hypotenuse.
  1. A central angle is 100°. What is an inscribed angle on the same arc?
  2. An inscribed angle is 35°. What is the central angle on the same arc?
  3. AB is a diameter and C is on the circle. Angle CAB is 25°. What is angle CBA?
  4. C and D are on the same arc. Angle ACB is 40°. Angle ADB is:
    • 40°
    • 80°
    • 20°
    • 140°
  5. The central angle measured around the major arc AB is 240°. What is an inscribed angle standing on that major arc?
  6. A quadrilateral has all four corners on a circle. One angle is 70°. What is the angle opposite it?
  7. In triangle OAB, O is the centre and the central angle AOB is 80°. What is angle OAB?
  8. C is on a circle and angle ACB = 90°. What can you say about AB?
    • it is a diameter
    • it is a tangent
    • it is half a radius
    • nothing can be said
  9. A right triangle has legs 6 cm and 8 cm. What is the radius of the circle through its three corners?
  10. A and B are the ends of a diameter. P is inside the circle, not on it. Angle APB is:
    • more than 90°
    • exactly 90°
    • less than 90°
    • always 45°
  11. C and D lie on opposite arcs of chord AB. Angle ACB is 40°. What is angle ADB?
Common mistakes
  • “The inscribed angle equals the central angle.” The inscribed angle is half. The vertex on the circle is further away, so the angle is narrower.
  • “The central angle is half the inscribed angle.” It is the other way round. A central angle of 100° goes with an inscribed angle of 50°. Picture the semicircle: the central angle is 180° and the inscribed angle 90°.
  • “The angle in a semicircle is 90° only when C is at the top.” Any point of the circle other than A and B gives 90°. Drag C along the circle in the trig lab: only the angles at A and B change.
Teacher's noteIn the 2022 Korean national curriculum the inscribed angle appears in middle school, grades 1 to 3, in a standard that asks learners to understand the properties of inscribed angles and justify them. Inscribed angle is on the term list. The curriculum also says that properties of circles and proportion are not covered, so results such as the intersecting chords theorem stay out. The step learners skip is the case split. They prove the theorem with the centre inside the angle and assume the rest. Draw three pictures side by side, with the centre inside the angle, on one of its sides, and outside it, and ask which step of the proof changes in each.
Key points
  • An inscribed angle is half the central angle on the same arc (Euclid III.20).
  • Inscribed angles on the same arc are equal (III.21).
  • The angle in a semicircle is 90° (III.31); conversely, a 90° inscribed angle stands on a diameter.
  • Inscribed angles on opposite arcs of a chord add to 180°.
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