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Home›Geometry and drawings›Circles

Pi and the circumference of a circle

Wrap 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.

Measure three round things

Find a mug, a plate and a drink can. For each one, wrap a strip of paper once around the rim, mark where it overlaps, then lay the strip flat and measure it. That length is the circumference, the distance around the circle. Next, measure the diameter straight across the top, through the middle. Suppose the mug comes out at 8 cm across and 25.1 cm around, and the plate at 24 cm across and 75.4 cm around. Divide the distance around by the distance across. For the mug, 25.1 ÷ 8 = 3.1375. For the plate, 75.4 ÷ 24 ≈ 3.1417. The plate is three times as wide as the mug, yet the two quotients agree to two decimal places.

That quotient is the same for every circle, small or large, and it has a name: pi, written π. OpenStax gives it as π = 3.141592654…, the ratio of the circumference to the diameter. Because the ratio never changes, two formulas follow at once. The circumference is C = πd, and since d = 2r, also C = 2πr. A circle 10 cm across has circumference 10π ≈ 31.4 cm. Your own measurements will never give π exactly. Paper stretches, rims have thickness, and a ruler only shows millimetres. Measuring gets you the first two digits. For more, you need an argument.

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A cylinder with radius 2 cm. Press Unfold: the curved side opens out into a rectangle. Its long side was wrapped around the lid, so its length is the lid’s circumference, 2πr ≈ 12.57 cm when r = 2. Change r and read the net line of the panel: the rectangle is always about 3.14 times as long as the lid is wide.

Squeezing π between two polygons

Archimedes of Syracuse found a way to trap π without measuring anything. Draw a regular hexagon inside a circle, with its six corners on the circle. The hexagon splits into six equilateral triangles that meet at the centre, so each side equals the radius and the perimeter is 6r = 3d. The circle bulges outside every side, so the circumference is longer than 3d: π > 3. Now draw a square around the circle, touching it at four points. Its side equals the diameter, so its perimeter is 4d, and the circle fits inside it: π < 4.

Blue: regular polygons inside the circle. Orange: regular polygons around it. For a circle of diameter 1, the inside perimeter is a lower bound for π and the outside perimeter an upper bound. Twelve sides already pin π between 3.106 and 3.215.

Doubling the number of sides tightens both bounds. For a circle of diameter 1, OpenStax lists the perimeters: with 8 sides, between 3.0615 and 3.3137; with 16 sides, between 3.1214 and 3.1826; with 64 sides, between 3.1403 and 3.1441. Archimedes began with hexagons and doubled four times, to 12, 24, 48 and 96 sides. He had no decimals and no trigonometry and worked out the square roots by hand as fractions. His conclusion was that π lies between 3 10/71 and 3 1/7, which in decimals is 3.1408 < π < 3.1429. With a calculator the 96-sided perimeters are 3.14103 and 3.14271. His two fractions sit just outside them, so both inequalities stay true.

SidesInside perimeterOutside perimeter
42.82844.0000
63.00003.4641
83.06153.3137
123.10583.2154
163.12143.1826
243.13263.1597
323.13653.1517
483.13943.1461
643.14033.1441
963.14103.1427
BackgroundThe symbol came much later than the number. MacTutor’s history of pi records that the Welsh mathematician William Jones was the first to use π with its present meaning, in 1706, writing “3.14159 etc. = π”. Euler adopted the symbol in 1737, and it quickly became standard. Calculations had gone on long before the letter. In China, Zu Chongzhi (430 to 501) gave the fraction 355/113, which is 3.1415929, correct to six decimal places. MacTutor notes that he is very unlikely to have known Archimedes’ work.

One more metre of string

Here is a question where the formula beats intuition. Wrap a string tightly around a ball, then add exactly 1 m to its length and lift it evenly away from the surface all the way round. How big is the gap? The new circumference is C + 1, so the new radius is (C + 1) ÷ 2π. That is the old radius plus 1 ÷ 2π, about 0.159 m. The gap is about 15.9 cm. The size of the ball never appears in the calculation, so the answer is the same for a tennis ball and for a ball as big as the Earth.

Using 3.14 or the π key on a calculator gives slightly different answers, because the key carries more digits. OpenStax Prealgebra makes the same remark. In the practice below, either answer is accepted.

C = πd = 2πr. π ≈ 3.14. Archimedes: 3 10/71 < π < 3 1/7.

  1. A circle has diameter 10 cm. Find its circumference (use 3.14 for π).
  2. A circle has radius 3 cm. Find its circumference (use 3.14 for π).
  3. A circle has circumference 47.1 cm. Find its diameter (use 3.14 for π).
  4. A circle has circumference 62.8 cm. Find its radius (use 3.14 for π).
  5. A wheel 0.7 m across turns 100 times without slipping. How far does it roll, in metres? (use 3.14 for π)
  6. A regular hexagon has its corners on a circle. Its perimeter equals:
    • 3 diameters
    • 2 diameters
    • 4 diameters
    • π diameters
  7. Which statement is true?
    • 3 10/71 < π < 3 1/7
    • π = 22/7 exactly
    • π = 3.14 exactly
    • π < 3
  8. Why does the polygon inside the circle give a lower bound for π?
    • each side is a shortcut across an arc
    • it has fewer sides than the circle
    • its corners are outside the circle
    • its sides are longer than the radius
  9. A string around a ball is made 1 m longer and lifted evenly. About how many centimetres is the gap? (one decimal place)
  10. Circumference ÷ diameter for a circle of radius 100 compared with a circle of radius 1:
    • the same
    • 100 times bigger
    • 100 times smaller
    • 10 times bigger
  11. A square is drawn around a circle of diameter 5 cm. What is the square’s perimeter?
Common mistakes
  • “π is 22/7.” 22/7 = 3.142857… while π = 3.141592…. Archimedes gave 22/7 as an upper bound, not as π itself.
  • “C = πr.” π compares the circumference with the diameter, not the radius: C = πd = 2πr. A quick check: the circumference must be more than 3 diameters, and πr is only about 1.57 diameters.
  • “My can gave 3.05, so π is different for small circles.” Small objects make measuring errors bigger in proportion. One millimetre wrong on a 6 cm can changes the quotient in the second digit. The ratio itself is the same for every circle.
Teacher's noteIn the 2022 Korean national curriculum this belongs to primary school, grades 5 to 6. One standard asks learners to measure the circumference and diameter of round objects, see that pi is a constant, and use an approximate value; the notes name 3.14. The next standard then has them find the circumference from the diameter without measuring. Where they stall is the class table of quotients: some come out at 3.05 or 3.2, and learners conclude that pi varies. Put everyone’s results on the board, sort them by object size, and ask which measurements are least trustworthy. The spread is error, and it shrinks for bigger objects.
Key points
  • Circumference ÷ diameter is the same for every circle: π = 3.14159…
  • C = πd = 2πr. Use 3.14 or the π key.
  • An inscribed hexagon gives π > 3; a square around the circle gives π < 4.
  • Archimedes, with 96-sided polygons: 3 10/71 < π < 3 1/7.
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