edubun
English
English한국어EspañolPortuguêsFrançaisDeutschItalianoNederlandsРусскийBahasa Indonesia日本語
Home›Geometry and drawings›Circles

Arc length and sector area

In 20 minutes the tip of a 10 cm minute hand sweeps 120°, a third of the clock face, and travels about 20.9 cm.

Try this first: the tip of a minute hand

A wall clock has a minute hand 10 cm long. How far does the tip travel between 3:00 and 3:20? Have a go before you read on. You need only what the earlier units gave you.

  1. A minute hand is 10 cm long. How far does its tip travel in 20 minutes? (use 3.14 for π, two decimal places)

Two facts settle it. The minute hand goes all the way round, 360°, in 60 minutes, so it turns 6° each minute and 120° in 20 minutes. A full turn would carry the tip around a circle of radius 10 cm, a distance of 2π × 10 ≈ 62.8 cm. Twenty minutes is 120 out of 360 degrees, a third of a turn, so the tip covers a third of that distance, about 20.9 cm. The path of the tip is an arc, and the region the hand sweeps over is a sector with a central angle of 120°.

Arc and area are the same fraction of the circle

The clock argument works for any sector. A sector with central angle θ takes θ/360 of a full turn, so its arc is θ/360 of the circumference: arc length ℓ = 2πr × θ/360. The same fraction applies to the area, since the sector is θ/360 of the whole disc: sector area S = πr² × θ/360. In a circle of radius 6 cm, a 60° sector is one sixth of the circle. Its arc is 2π × 6 ÷ 6 = 2π ≈ 6.28 cm and its area is π × 36 ÷ 6 = 6π ≈ 18.84 cm². Going the other way, an arc of 5π cm in a circle of radius 10 cm is 5π ÷ 20π = 1/4 of the circumference, so its central angle is 90°.

Arcs and sector areas are proportional to the central angle: double the angle and both double. Chords are not. In a circle of radius 10, a 60° central angle cuts off a chord of length 10, because the triangle with two radii is equilateral. A 120° angle cuts off a chord of about 17.3, not 20. Meanwhile the arc doubles from about 10.5 to about 20.9. The straight chord cuts across the bend, and the wider the angle, the more it saves. Equal angles do still give equal chords: Euclid proves in Book III, Proposition 28 that in equal circles equal chords cut off equal arcs.

Left: a sector with central angle θ, radius r and arc ℓ. Right: in a circle of radius 10, the orange 60° angle has a chord of 10 and an arc of about 10.5; the blue 120° angle has twice the arc, about 20.9, but a chord of only about 17.3.

A sector area that needs no angle

Multiply the arc length by half the radius and see what comes out: ½ × r × ℓ = ½ × r × 2πr × θ/360 = πr² × θ/360. That is the sector area again, so S = ½rℓ. It looks like the triangle formula, half of base times height, and for a good reason: a thin sector is almost a triangle whose base is the arc and whose height is the radius. A sector of radius 8 cm with an arc of 10 cm has area ½ × 8 × 10 = 40 cm², and you never needed the angle. This is the formula behind the curved side of a cone.

Loading the 3D model

Drag to turn. Scroll or pinch to zoom.

A cone with r = 3 cm and h = 4 cm, so the slant height l is 5 cm. Press Unfold: the curved side flattens into a sector of radius 5 cm. Its arc used to wrap around the base, so the arc is 2π × 3 cm long, and the central angle is 360° × 3/5 = 216°. The side area is ½ × 5 × 6π = 15π ≈ 47.12 cm², the πrl in the panel.

Every pie chart is made of sectors

A pie chart is a circle divided into sectors whose central angles are shares of 360°. A category that makes up 25% of the total gets a slice with a central angle of 0.25 × 360° = 90°. In the chart below, 16 of 50 students bought 3 books: 16 ÷ 50 = 32%, and 0.32 × 360° = 115.2°. Because area is proportional to the angle, a slice’s area is the same share of the whole as its count.

Loading

Drag to turn. Scroll or pinch to zoom.

Click a slice. The panel shows its share in percent and its central angle in degrees. The six angles add up to 360°.
A segment (the region between a chord and its arc) is a sector minus a triangle. In a circle of radius 10 cm, the 90° segment is 25π − 50 ≈ 28.54 cm²: the quarter circle minus the right triangle with legs 10 and 10.

Arc ℓ = 2πr × θ/360. Area S = πr² × θ/360 = ½ r ℓ.

  1. Radius 6 cm, central angle 60°. Find the arc length (use 3.14 for π).
  2. Radius 6 cm, central angle 60°. Find the sector area (use 3.14 for π).
  3. A circle has radius 10 cm. An arc is 5π cm long. What is its central angle?
  4. A sector has radius 8 cm and arc length 10 cm. Find its area.
  5. In a pie chart, a category is 25% of the total. What is its central angle?
  6. A pie chart slice has a central angle of 72°. What fraction of the total is it?
    • 1/5
    • 1/4
    • 1/6
    • 72/100
  7. A cone has base radius 3 cm and slant height 5 cm. Unfolded, its side is a sector. What is the central angle?
  8. In one circle you double a central angle. Which quantities double?
    • the arc and the sector area
    • the arc, the area and the chord
    • only the chord
    • nothing doubles
  9. A 30 cm pizza is cut into 8 equal slices. How long is the crust edge of one slice? (use 3.14 for π)
  10. Find the area of the segment cut off by a 90° central angle in a circle of radius 10 cm (use 3.14 for π).
  11. Through how many degrees does a minute hand turn in 15 minutes?
Common mistakes
  • “Arc length = πr² × θ/360.” That is the area formula. Arc length comes from the circumference 2πr and is measured in cm; area comes from πr² and is measured in cm².
  • “Double the central angle, double the chord.” Only arcs and areas are proportional to the angle. In a circle of radius 10 the chords for 60° and 120° are 10 and about 17.3.
  • “A 45° sector of a circle of diameter 8 has arc 2π × 8 × 45/360.” The formula uses the radius. With r = 4 the arc is 2π × 4 × 45/360 = π ≈ 3.14, half of the wrong answer.
Teacher's noteIn the 2022 Korean national curriculum, one standard for middle school, grades 1 to 3, asks learners to understand the relationship between the central angle and the arc of a sector and to use it to find arc length and area. The curriculum also says that complicated composite figures are not covered, so keep practice to sectors, segments and simple combinations. The common stall is memorising the two formulas as separate facts. Start from a clock face or a pizza and ask “what fraction of the circle is this?” Learners who say “a third” before writing anything rarely mix up the formulas afterwards.
Key points
  • A sector with central angle θ is θ/360 of the circle.
  • Arc ℓ = 2πr × θ/360. Area S = πr² × θ/360 = ½ r ℓ.
  • Arc and area are proportional to the central angle; chord length is not.
  • Pie charts: angle = share × 360°. Segment = sector − triangle.
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