11. Areas Related to Circles

4 topics49 min

Chapter 11 of NCERT Mathematics for Class 10: Areas Related to Circles. Two sections — Naming the pieces and Measuring the pieces. Four videos, 49 min in all.

Worked answers to the 14 exercise questions in this chapter: Exercise 11.1

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Part 1

Naming the pieces

9 minSector versus segment, and which one major and minor refer toTwo of these words describe the CUT and two describe which piece you kept. Either cut leaves exactly two pieces - never one, never three - so neither sector nor segment names a region at all. Each names a pair, and that is why the major piece is always the whole disc take away the minor one.9 min · 12 parts

Part 2

Measuring the pieces

12 minA sector's area as its share of the full turnThe rule for a sector's area is one old fact plus one claim that almost nobody states: that area changes in strict proportion to the angle. The claim is false one step away - the piece past a chord does not scale at all - so something has to be true of the wedge that is not true of its neighbour. Two things are, and even they are not quite enough.12 min · 14 parts13 minThe arc's length by the same share argumentThe rule for the length of an arc is the sector-area argument run a second time on a different quantity - and the point of it is that the ARGUMENT transfers, not the answer. That argument runs on exactly two facts, and both of them have to be checked again before they can be used, because each one has a shape that fails it.13 min · 14 parts14 minA segment as what is left when the triangle is taken awayHalve the angle and the wedge halves. Halve the angle and the piece past the chord does not - it drops to less than a seventh. That one collapse is why the share-of-the-turn argument stops dead here, and what replaces it is a subtraction that is exact, not approximate.14 min · 14 parts

What you will be able to do

  • Distinguish a sector from a segment by naming what bounds each one
  • State what makes area proportional to the sector's angle, in terms of turning and of adding
  • State the two properties that let the share argument be re-used, and check each against arc length rather than assuming it
  • Explain why the sector argument cannot be re-run for a segment

What this chapter assumes you already know

  • A circle as the points at a fixed distance from a centre, and the words centre, radius, chord and arc, all carried in from Class IX and assumed by name here
  • That a chord joins two points of the circle and cuts the circle's curve into two arcs
  • Area of a circle as πr², used from p. 155 onward and needed here only as the size of the whole
  • That the angles round a point add to 360°
  • Area of a circle as πr², and the standard approximations π ≈ 22/7 and π ≈ 3.14
  • The unitary method as a general habit: get to one, then scale up
  • That a full turn is 360°, and that angles at a point add
  • Confidence with fractions of the form θ/360 and with cancelling before multiplying

It builds directly on Counting shared points sorts every line into one of three kinds.

Where people usually slip up

Sentences students actually say, taken from the notes the videos were made from. Each one is answered on the page of the video it belongs to.

  • "A segment is the same as a sector, roughly."
  • "The shaded region is the sector."
  • "Minor means small, so a minor sector is always a thin sliver."
  • "Major and minor are decided by the angle you were given."
  • "A chord and a diameter behave the same way."
  • "Sector always means the small one."
  • "The letters in OAPB are decoration."
  • "The formula is just something to memorise."