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Home›Geometry and drawings

Circles

Six units on circles: the parts of a circle and how to find its centre, pi and the circumference with Archimedes’ polygon bounds, the area of a circle from rearranged sectors, arc length and sector area as fractions of 360°, central and inscribed angles with the angle in a semicircle, and tangents, where the radius meets the tangent at a right angle and two tangents from one point are equal. Drag a point round a circle, unroll a cylinder, unfold a cone and click the slices of a pie chart.

Progress

About this course

Euclid's definition of a circle, Book I Definition 15 in the D. E. Joyce edition, describes a plane figure bounded by one line such that all straight lines falling on it from one point inside are equal. Joyce's notes point out two differences from today. For Euclid the circle is the flat region, and the curved line is the circumference. And Euclid had no separate word for radius at all: he wrote "the line from the centre". 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.

The six units start with the parts of a circle, where two chords on a plate have perpendicular bisectors that cross at exactly one point, the centre. Pi and the circumference follow, with Archimedes pinning pi between 3 10/71 and 3 1/7. MacTutor records that William Jones was the first to use π in its present meaning, in 1706, and Zu Chongzhi (430 to 501) gave 355/113, correct to six decimal places. The area of a circle compares a 30 cm pizza with a 20 cm one: 1.5 times as wide, 2.25 times as much to eat. Arc length and sector area use a 10 cm minute hand, which in 20 minutes sweeps 120°. Central and inscribed angles include Euclid's Book III Proposition 31, that the angle in a semicircle is 90°. The last unit shows that from a point 13 cm from the centre of a circle of radius 5 cm, both tangents are exactly 12 cm long. In the labs you drag a point round a circle, unroll a cylinder, unfold a cone and click the slices of a pie chart.

Before you start

Fractions of a full turn help when finding arc lengths and sector areas. Angles also helps with the central-angle, inscribed-angle and tangent arguments. The opening lesson identifies the parts of a circle, so you can keep radius, diameter, chord and tangent distinct before using formulas.

When you finish

You can find a circle’s centre from chords, distinguish radius from diameter, and calculate circumference and area. You can use a fraction of 360° to find an arc length or sector area, relate central and inscribed angles, recognise the right angle in a semicircle, and use equal tangents from the same outside point.

  1. 1Parts of a circle: radius, diameter, chord, arc, sector and tangentDraw 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.
  2. 2Pi and the circumference of a circleWrap a string once round anything circular and it comes out a little more than three diameters long. Archimedes pinned that “little more” between 3 10/71 and 3 1/7.
  3. 3Area of a circle: why it is πr²A pizza 30 cm across is 1.5 times as wide as a 20 cm pizza, but it has 2.25 times as much to eat.
  4. 4Arc length and sector areaIn 20 minutes the tip of a 10 cm minute hand sweeps 120°, a third of the clock face, and travels about 20.9 cm.
  5. 5Central and inscribed anglesJoin 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.
  6. 6Tangents: the right angle at the point of contact and equal tangent lengthsFrom a point 13 cm from the centre of a circle of radius 5 cm, both tangents are exactly 12 cm long.

Related courses

  • Angles
    Five units on angles: measuring them with a protractor, naming the types, the pairs that add to 90° and 180°, the eight angles where a line crosses two parallel lines, and the angle sums of triangles and polygons. The trig lab on each page has a protractor, a pair of parallel lines and a triangle whose corners you can drag.
  • Triangles
    Seven units on triangles: sorting them by sides and angles, the 180° angle sum and the exterior angle, which three lengths can make a triangle, the congruence conditions and why side-side-angle fails, similar triangles and scale, the Pythagorean theorem with two proofs and its converse, and area from base and height or from three sides with Heron’s formula. Drag the corners of a triangle in the trig lab and watch the numbers change.
  • Area and volume of solids
    A cereal box, a tin can, an ice cream cone and a ball each have a formula for how much card covers them and how much they hold. Unfold each solid in 3D, change its size with sliders and watch the numbers follow.
  • Trigonometry
    Six units of right-triangle and unit-circle trigonometry. You name the sides of a right triangle, define sine, cosine and tangent as ratios, learn the exact values for 30°, 45° and 60°, solve triangles from one side and one angle, measure heights you cannot reach, switch to radians and the unit circle, and draw the sine wave. The trig lab has a resizable right triangle, a height-measuring range and a unit circle tied to its graph.
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