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Home›Geometry and drawings›Circles

Area of a circle: why it is πr²

A 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.

The “one and a half times” guess

A pizza shop sells a 20 cm pizza and a 30 cm pizza. The bigger one is 1.5 times as wide, and most people guess it holds about 1.5 times as much food. Work it out instead. The 20 cm pizza has radius 10 cm and area π × 10² ≈ 314 cm². The 30 cm pizza has radius 15 cm and area π × 15² ≈ 707 cm². The ratio is 225 ÷ 100 = 2.25. The guess fails because area grows in two directions at once: stretch every length by 1.5 and the area grows by 1.5 × 1.5. Before trusting the formula A = πr² that produced these numbers, it is worth seeing where it comes from.

Cut the circle into slices and lay them out

Cut a circle into 8 equal sectors, as you would a pizza. Lay them side by side, alternately pointing up and pointing down. The result looks like a bumpy parallelogram. Its height is about r, the radius. Along the top run the arcs of four sectors and along the bottom the arcs of the other four, so each wavy edge is half the circumference, πr. Now cut 24 sectors instead. The bumps get smaller, the slanted ends get steeper, and the shape gets closer to a rectangle of width πr and height r. With more and more sectors the shape comes as close to that rectangle as you like, so the circle’s area is πr × r = πr².

Left: a circle cut into 8 sectors, the top half blue and the bottom half orange. Right: the same sectors laid alternately, first 8, then 24. The shape approaches a rectangle with height r and width πr, half the circumference.

Check the answer against squares. A square drawn around a circle of radius r has side 2r and area 4r². A square drawn inside, with its corners on the circle, has diagonals of length 2r and area 2r². The circle sits between them, so its area must be between 2r² and 4r², and πr² ≈ 3.14r² fits. Regular polygons inside the circle close the gap the same way they did for the circumference in unit 2: a regular 16-gon inside covers about 3.0615r², a 64-gon about 3.1365r².

Most errors come from the input

The formula is short, so the mistakes happen before it. If you are given the diameter, halve it first: a circle 8 cm across has r = 4 and area 16π ≈ 50.24 cm². If you are given the circumference, find r from C = 2πr first: C = 31.4 cm gives r = 5 cm and area 25π ≈ 78.5 cm². Areas come in square units, cm² or m². And because r is squared, doubling the radius multiplies the area by 4, and tripling it multiplies the area by 9.

Shapes made of rectangles and half circles work the same way, piece by piece. OpenStax describes a running track shaped like a rectangle 105 m long and 68 m wide with a half circle on each end. The two half circles have diameter 68 m and together make one whole circle of radius 34 m. The enclosed area is 105 × 68 + π × 34² = 7,140 + 3,631.68… ≈ 10,771.68 m². A ring between two circles is the big circle minus the small one: between radii 5 cm and 3 cm the ring covers π(25 − 9) = 16π ≈ 50.24 cm².

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Archimedes showed that the surface of a sphere is four times the area of a great circle, the circle you get by cutting the sphere through its centre. The panel gives the sphere’s area as 4πr². For r = 3 cm that is about 113.1 cm², four times π × 3² ≈ 28.27 cm². Press Archimedes cylinder to see the cylinder that just holds the sphere.
BackgroundAn older answer survives from Egypt. MacTutor’s history of pi notes that the Rhind papyrus, dated to about 1650 BC, gives good evidence for the value 4 × (8/9)² ≈ 3.16. That number is what you get if you replace a circle by a square whose side is 8/9 of the diameter. For a circle 9 units across, the square has side 8 and area 64, while the circle’s true area is π × 4.5² ≈ 63.6. The square is less than 1% too big, a very good result for a rule that needs no π at all.

A = πr² = (half the circumference) × r. Double r and the area is 4 times as big.

  1. Find the area of a circle with radius 10 cm (use 3.14 for π).
  2. Find the area of a circle with diameter 8 cm (use 3.14 for π).
  3. How many times as much area does a 30 cm pizza have as a 20 cm pizza?
  4. You double the radius of a circle. Its area becomes:
    • 4 times as big
    • 2 times as big
    • 8 times as big
    • π times as big
  5. A circle has circumference 31.4 cm. Find its area (use 3.14 for π).
  6. Find the area of a half circle with radius 6 cm (use 3.14 for π).
  7. A ring lies between circles of radius 5 cm and 3 cm with the same centre. Find its area (use 3.14 for π).
  8. A circle has area 28.26 cm². Find its radius (use 3.14 for π).
  9. When the sectors are laid out alternately, the long side of the near-rectangle is:
    • half the circumference
    • the whole circumference
    • the diameter
    • the radius
  10. The area of a circle of radius r is between:
    • 2r² and 4r²
    • r² and 2r²
    • 4r² and 8r²
    • 2r and 4r
  11. A track is a 105 m by 68 m rectangle with a half circle on each 68 m end. Find the enclosed area to the nearest square metre (use the π key).
Common mistakes
  • “Area of a circle = 2πr.” That is the circumference, a length in cm. Area needs r twice and comes in cm². For r = 10 the circumference is about 62.8 cm and the area about 314 cm².
  • “With the diameter, area = π × d².” Using d in place of r makes the answer 4 times too big, because d² = 4r². Halve the diameter first.
  • “1.5 times as wide means 1.5 times the area.” Area scales by the square of the length factor: 1.5² = 2.25. The same rule holds for every shape, not only circles.
Teacher's noteThe 2022 Korean national curriculum places the area of a circle in primary school, grades 5 to 6. The notes to that standard say learners should understand how to find the area of a circle by using the method for the area of a rectangle, which is the sector rearrangement on this page. Learners usually accept the 8-slice picture but doubt the step to “exactly a rectangle”. Let them cut paper circles into 8 and then 16 slices and compare the edges. The bumps visibly flatten, and the question becomes how far the process can go, which is the right question.
Key points
  • A = πr². With 3.14: r × r × 3.14.
  • Cut into sectors and laid alternately, a circle becomes nearly a rectangle πr wide and r high.
  • Given the diameter, halve it first. Given the circumference, find r first.
  • Doubling r multiplies the area by 4.
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