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  1. Learning objectives
  2. Core idea
  3. Key formulae
  4. Worked examples
  5. Explore the graph
  6. Common mistakes
  7. Exam tip
  8. Guided practice
  9. Practice questions
  10. Challenge
  11. Summary
  12. Quick quiz
  13. Same topic, different board
  14. Past paper questions
Learn9231Paper 1 · Further Pure Mathematics 1Polar coordinatesPolar curves and areas
Polar coordinates · Subtopic 1 of 1

Polar Coordinates

A point is given by its distance rr from the pole and the angle θ\theta from the initial line: x=rcos⁡θx=r\cos\theta, y=rsin⁡θy=r\sin\theta. Curves r=f(θ)r=f(\theta) draw spirals, cardioids and roses. The area swept out is 12∫r2 dθ\frac12\int r^2\,d\theta.

Cambridge 9231 Paper 1Edexcel YFM01 WFM02Edexcel 9FM0 Paper 1Edexcel 9FM0 Paper 2AQA 7367 Paper 1AQA 7367 Paper 2≈ 45 minRevision 1 · updated 2026-10-27
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Section 1 of 14

Learning objectives

  • Convert between polar and Cartesian coordinates and equations.
  • Sketch curves such as circles, cardioids, spirals and roses.
  • Find areas using 12∫r2 dθ\frac12\int r^2\,d\theta.
  • Find tangents parallel or perpendicular to the initial line.
Pause and think

A lighthouse beam sweeps round. How would you describe where a ship is, using the lighthouse?

Section 2 of 14

Core idea

Start hereDescribing a spiral

Try writing the equation of a spiral as y=f(x)y=f(x): impossible, because it crosses each vertical line many times. But "the further round you go, the further out you get" is simply r=θr=\theta. Some curves are natural in polar coordinates.

The big idea

(r,θ)(r,\theta): distance from the pole and angle from the initial line. x=rcos⁡θx=r\cos\theta, y=rsin⁡θy=r\sin\theta, r2=x2+y2r^2=x^2+y^2. Area =12∫αβr2 dθ=\frac12\int_\alpha^\beta r^2\,d\theta.

1. Converting

Worked exampler=2cos⁡θr=2\cos\theta

Write in Cartesian form and identify the curve.

Goal
Equation.
Strategy
Multiply by rr to create r2r^2 and rcos⁡θr\cos\theta.
  1. r2=2rcos⁡θr^2=2r\cos\theta
    Multiply both sides by rr.
  2. x2+y2=2xx^2+y^2=2x
    Substitute.
  3. (x−1)2+y2=1(x-1)^2+y^2=1
    Complete the square.
AnswerCircle centre (1,0)(1,0), radius 11, passing through the pole.
✓ Checkθ=0\theta=0: r=2r=2, the point (2,0)(2,0) is on the circle ✓.

2. Areas

Why 12∫r2 dθ\frac12\int r^2\,d\theta
  1. δA≈12r2 δθ\delta A\approx\tfrac12r^2\,\delta\theta
    A thin slice is nearly a sector of radius rr and angle δθ\delta\theta.
  2. A≈∑12r2 δθA\approx\sum\tfrac12r^2\,\delta\theta
    Add the slices.
  3. A=∫αβ12r2 dθA=\int_\alpha^\beta\tfrac12r^2\,d\theta
    Limit as δθ→0\delta\theta\to0.
Worked exampleCardioid r=2(1+cos⁡θ)r=2(1+\cos\theta)

Find the enclosed area.

Goal
Area.
Strategy
Expand, use cos⁡2θ=12(1+cos⁡2θ)\cos^2\theta=\frac12(1+\cos2\theta).
  1. A=12∫02π4(1+cos⁡θ)2 dθA=\tfrac12\int_0^{2\pi}4(1+\cos\theta)^2\,d\theta
    Full turn.
  2. =2∫02π(32+2cos⁡θ+12cos⁡2θ)dθ=2\int_0^{2\pi}\left(\tfrac32+2\cos\theta+\tfrac12\cos2\theta\right)d\theta
    Expand and use the double-angle identity.
  3. =2[32θ+2sin⁡θ+14sin⁡2θ]02π=6π=2\left[\tfrac32\theta+2\sin\theta+\tfrac14\sin2\theta\right]_0^{2\pi}=6\pi
    Sine terms vanish at both limits.
Answer6π6\pi.
✓ CheckGeneral cardioid area 32πa2\frac32\pi a^2 with a=2a=2 ✓.

3. Tangents

Worked exampleTangents to r=1+cos⁡θr=1+\cos\theta parallel to the initial line

Find θ\theta in (0,π)(0,\pi).

Goal
θ\theta.
Strategy
Horizontal tangent: dydθ=0\frac{dy}{d\theta}=0 with y=rsin⁡θy=r\sin\theta.
  1. y=sin⁡θ+sin⁡θcos⁡θ=sin⁡θ+12sin⁡2θy=\sin\theta+\sin\theta\cos\theta=\sin\theta+\tfrac12\sin2\theta
    y=rsin⁡θy=r\sin\theta.
  2. dydθ=cos⁡θ+cos⁡2θ=2cos⁡2θ+cos⁡θ−1\frac{dy}{d\theta}=\cos\theta+\cos2\theta=2\cos^2\theta+\cos\theta-1
    Differentiate; cos⁡2θ=2cos⁡2θ−1\cos2\theta=2\cos^2\theta-1.
  3. (2cos⁡θ−1)(cos⁡θ+1)=0(2\cos\theta-1)(\cos\theta+1)=0
    Factorise.
Answerθ=π3\theta=\frac\pi3 (and θ=π\theta=\pi is the pole).
✓ CheckAt θ=π3\theta=\frac\pi3, r=32r=\frac32, y=334≈1.30y=\frac{3\sqrt3}4\approx1.30, the top of the curve ✓.
Common misconceptionMissing the half Area is 12∫r2 dθ\frac12\int r^2\,d\theta, not ∫r dθ\int r\,d\theta.
Edge caseNegative r Most syllabuses restrict to r≥0r\ge0: leave out angles where f(θ)<0f(\theta)<0 when sketching.

Summary

  • x=rcos⁡θx=r\cos\theta, y=rsin⁡θy=r\sin\theta.
  • Area 12∫r2dθ\frac12\int r^2d\theta.
  • Tangents via rsin⁡θr\sin\theta or rcos⁡θr\cos\theta.
Key words
Pole
The origin in polar coordinates.
Initial line
The half-line θ=0\theta=0.
HistoryBernoulli and Newton. Jacob Bernoulli used polar coordinates for spirals in 1691; Newton had listed them earlier as one of several coordinate systems.
Fun factSunflowers. Seeds in a sunflower head sit at r=cnr=c\sqrt n, θ=n×137.5∘\theta=n\times137.5^\circ: a polar pattern that packs them perfectly.
Section 3 of 14

Key formulae

Conversionx=rcos⁡θ, y=rsin⁡θ, r2=x2+y2, tan⁡θ=yxx=r\cos\theta,\ y=r\sin\theta,\ r^2=x^2+y^2,\ \tan\theta=\frac yx
Area of a sector-like regionA=12∫αβr2 dθA=\frac12\int_\alpha^\beta r^2\,d\theta
Tangentsparallel to initial line: ddθ(rsin⁡θ)=0;perpendicular: ddθ(rcos⁡θ)=0\text{parallel to initial line: }\frac{d}{d\theta}(r\sin\theta)=0;\quad\text{perpendicular: }\frac{d}{d\theta}(r\cos\theta)=0
TrickTable first. For a sketch, tabulate rr at θ=0,π4,π2,…\theta=0,\frac\pi4,\frac\pi2,\ldots. Where r=0r=0 the curve passes through the pole, and the line at that angle is a tangent there.
Section 4 of 14

Worked examples

Example 1 · Cardioid area0 / 3 steps
Find the area enclosed by r=2(1+cos⁡θ)r=2(1+\cos\theta).
    Example 2 · Convert0 / 2 steps
    Write r=2cos⁡θr=2\cos\theta in Cartesian form.
      Section 5 of 14

      Explore the graph

      Tracing a cardioid
      Loading the interactive graph…
      Try this
      1. Slide ss (the angle θ\theta) from 00 to 2π2\pi and watch the point travel round r=a(1+cos⁡θ)r=a(1+\cos\theta).
      2. At s=πs=\pi the point reaches the pole: r=0r=0.

      The point (rcos⁡θ,rsin⁡θ)(r\cos\theta,r\sin\theta) with r=a(1+cos⁡θ)r=a(1+\cos\theta).

      In the real worldRadar screens. Radar displays plot contacts by range and bearing, which is exactly polar coordinates, before any conversion to a map grid.
      Section 6 of 14

      Common mistakes

      IncorrectArea =∫r dθ=\int r\,d\theta
      Correct12∫r2 dθ\frac12\int r^2\,d\theta
      A thin sector has area 12r2δθ\frac12r^2\delta\theta.
      IncorrectUsing negative rr values
      CorrectMost specs take r≥0r\ge0
      Exclude angles where r<0r<0 when sketching.
      IncorrectIntegrating cos⁡2θ\cos^2\theta directly
      CorrectUse 12(1+cos⁡2θ)\frac12(1+\cos2\theta)
      There's no direct antiderivative.

      Cartesian: x and y

      Polar: hold my spiral.

      Section 7 of 14

      Exam tip

      Exam tipCheck symmetry. If rr depends only on cos⁡θ\cos\theta, the curve is symmetric in the initial line: integrate from 00 to π\pi and double.
      Section 8 of 14

      Guided practice

      SA-FP-POL-G011 markL1 FoundationNon-calculatorConverting
      The point has polar coordinates (2,π3)\left(2,\frac\pi3\right). Find its xx-coordinate.

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      Salam Academy Original Practice · Proposed mark scheme
      Section 9 of 14

      Practice questions

      SA-FP-POL-0012 marksL2 StandardCalculatorAreas
      Use 12∫r2 dθ\frac12\int r^2\,d\theta to find the area enclosed by r=3r=3 (to 3 s.f.).

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      Salam Academy Original Practice · Proposed mark scheme
      SA-FP-POL-0023 marksL3 ExamCalculatorAreas
      Find the area enclosed by the cardioid r=2(1+cos⁡θ)r=2(1+\cos\theta) (to 3 s.f.).

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      Salam Academy Original Practice · Proposed mark scheme
      SA-FP-POL-0032 marksL2 StandardNon-calculatorConverting
      The curve r=2cos⁡θr=2\cos\theta is a circle. Find its radius.

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      Salam Academy Original Practice · Proposed mark scheme
      SA-FP-POL-0043 marksL3 ExamCalculatorAreas
      Find the area swept by r=θr=\theta from θ=0\theta=0 to θ=π\theta=\pi (to 3 s.f.).

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      Salam Academy Original Practice · Proposed mark scheme

      More questions on every skill in this subtopic are in the question bank under Practise.

      Section 10 of 14

      Challenge

      SA-FP-POL-0054 marksL4 ChallengingNon-calculatorTangents
      Find the value of θ\theta, in degrees with 0<θ<1800<\theta<180, where the tangent to r=1+cos⁡θr=1+\cos\theta is parallel to the initial line.

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      Salam Academy Original Practice · Proposed mark scheme
      Section 11 of 14

      Summary

      • x=rcos⁡θx=r\cos\theta, y=rsin⁡θy=r\sin\theta.
      • Area 12∫r2dθ\frac12\int r^2d\theta.
      • Tangents: differentiate rsin⁡θr\sin\theta or rcos⁡θr\cos\theta.
      Section 12 of 14

      Quick quiz

      Q1 · SA-FP-POL-Q011 markL1 FoundationNon-calculatorSketching
      The polar curve r=3r=3 is
      Choose one option

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      Salam Academy Original Practice · Proposed mark scheme
      Section 13 of 14

      Same topic, different board

      Specification
      9231 AS & A Level Further Mathematics
      Where
      Paper 1 · 1.5 Polar coordinates
      Expected
      Convert between polar and Cartesian; sketch curves r = f(θ); area ½∫r² dθ.
      Calculator
      Permitted on all papers; exact answers are expected where the question asks.
      Typical style
      Area questions often need a double-angle identity before integrating.
      Open in
      9231 · Paper 1
      Section 14 of 14

      Past paper questions

      No past paper questions are tagged to this subtopic yet. We add papers every week.

      Browse past papers

      Salam Academy original content · Content status: reviewed and published. Syllabus mapping: Cambridge International 9231, Current syllabus (official specification).