Describing a point by how far out and at what angle, rather than across and up. Curves that are awkward in Cartesian form often become simple.
Syllabus points covered
- Understand the relationship between Cartesian and polar coordinates and convert between them
- Sketch simple polar curves for a given range of θ
- Find the area of a sector of a polar curve using the formula ½∫r² dθ
- Find the maximum value of r, and the points on a curve where x or y is greatest
Common mistakes
- Forgetting the ½ in the polar area formula, or squaring after integrating rather than before.
- Using the wrong limits — they must correspond to the part of the curve being described.
- Getting the angle in the wrong quadrant when converting from Cartesian coordinates.
- Sketching a curve for a range of θ where r is negative without handling it correctly.
Trouble sketching polar curves?
Print this out and sketch straight onto it. Working on a ready-made radial grid takes the guesswork out of plotting r against θ, and lets you concentrate on the shape of the curve rather than on drawing accurate circles. Once the grid feels familiar you will find you can sketch these curves on plain paper without it.
Worksheets
Use these as soon as you've learned the topic and need to practise it. Start with the worksheet, check yourself against the answers, and only then look at the step-by-step solutions.
- Practice worksheet
- Worksheet answers
- Step-by-step solutionsComing soon
Topical past papers
Move on to these once you've worked through the worksheets and are ready for harder, exam-style questions. Real past-paper questions on this topic, with the official mark scheme and full worked solutions.
- Topical questions
- Mark scheme
- Step-by-step worked solutionsComing soon
Looking for whole papers by session? All Further Mathematics past papers