PrepShorts · Study sheet · Class 10 Mathematics · Chapter 11, Areas Related to Circles
Chapter 11 · Areas Related to Circles
Sector versus segment, and which one major and minor refer to
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Two 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.
The idea
Two of this page's words describe the cut and two describe which piece you kept. Draw two radii and the disc falls apart one way; draw a chord and it falls apart another way — and either cut, being a single cut right across a filled circle, always leaves exactly two pieces. So neither word names one region: each names a pair, and major and minor are what pick a member out of the pair. That is not a labelling nicety. Because the pair leaves nothing over and nothing double-counted, the larger piece is always the whole disc minus the smaller one, and every "find the major …" instruction in this chapter is that subtraction wearing a name. The chapter's Remark, which lets a bare sector or segment stand for the minor one, is a convention adopted for brevity rather than a fact about circles — and the surprise is how little of the chapter actually leans on it. Both boxed results on p. 155 are stated for a named angle and hold at any angle, which is why p. 156 can feed 330 straight into the sector rule to get a major sector; the segment relation names its regions by letter. Exactly one printed result rides on the convention: Summary item 3 on p. 160, which names no region at all and would be false read of a major segment.
What you should be able to do
- Distinguish a sector from a segment by naming what bounds each one
- Point to both sectors, or both segments, produced by a single cut, and say why there are never more or fewer than two
- Apply major and minor correctly, and say what decides which is which
- Compute the angle of the major sector from the angle of the minor one
- State the major piece as the whole disc minus the minor piece, and justify the subtraction from the fact that the two pieces tile the disc
- Read the chapter's Remark and say what a bare "segment" will mean for the rest of the chapter
- Identify, in a question that names both a segment and a sector, which region each phrase points at
- Say which case the major/minor naming does not settle, and why
Words to know
| Term | Definition in one line | First introduced |
|---|---|---|
| sector | the piece of a filled circle bounded by two radii and the arc joining their outer ends | printed in §11.1, p. 154, and recalled there as vocabulary from earlier classes |
| segment | the piece of a filled circle bounded by a chord and one of the two arcs that chord cuts off | printed in §11.1, p. 154 |
| minor sector | the smaller of the two sectors a given pair of radii produces | printed in §11.1, p. 154, and also lettered inside the artwork of Fig. 11.1 |
| major sector | the larger of that same pair | printed in §11.1, p. 154, and lettered inside Fig. 11.1 |
| minor segment | the smaller of the two segments a given chord produces | printed in §11.1, p. 154, and lettered inside Fig. 11.2 |
| major segment | the larger of that same pair | printed in §11.1, p. 154, and lettered inside Fig. 11.2 |
| angle of the sector | the angle the two bounding radii open out at the centre | printed in §11.1, p. 154, and used again on p. 158 in Exercise 11.1 question 1 |
| chord | a segment with both ends on the circle; here, the straight edge of a segment | used on p. 154 as prior knowledge, and again in four of Exercise 11.1's fourteen questions — 4, 5 (iii), 6 and 7, all on p. 158 |
| disc | the filled circular region rather than the curve bounding it | printed once, on p. 155, as a gloss on "circular region"; the chapter otherwise says circular region |
| central angle | the angle a chord or an arc opens out at the centre, whatever region is being named | an added term; the chapter says "angle of the sector" for a sector and, for a chord, describes the angle it makes at the centre |
Where people slip up
- "A segment is the same as a sector, roughly." They coincide in exactly one case and differ in every other. One is bounded by two straight edges and an arc, the other by one straight edge and an arc — so at a straight angle, where the two radii lie along the chord and the triangle between them flattens to nothing, the two names pick out the same semicircular region. At every angle strictly between 0° and 180° the two regions differ, and what separates them is that triangle on the two radii and the chord. Note that this is a statement about one cut read two ways; Figs. 11.1 and 11.2 are not that pair, since the wedge in the first opens about 60° and the chord in the second is drawn on a quite different cut. The triangle relation is the whole content of A segment as what is left when the triangle is taken away.
- "The shaded region is the sector." Shading is the printer's pointer, not the definition. In both figures the unshaded region is equally a sector, or equally a segment; the chapter says so explicitly, and a student who reads shading as definitional will not know what the major piece is.
- "Minor means small, so a minor sector is always a thin sliver." Minor means smaller than its partner. A sector of 179° is still the minor one.
- "Major and minor are decided by the angle you were given." They are decided by size. Being handed a 240° angle does not make that sector minor; it makes it the major one, and its partner is the 120° minor sector.
- "A chord and a diameter behave the same way." A chord through the centre makes two pieces of equal area, so neither is major or minor and the chapter's naming has nothing to say. Say this out loud rather than leaving it as a hole: the same happens for two radii opening out to a straight angle.
- "Sector always means the small one." Only under the Remark on p. 154, and only inside this chapter. Read the words in a physics or a mensuration question and the convention may not travel with them.
- "The letters in OAPB are decoration." They are a route: out from O to A, round the arc through P, back to B and in to O. Reading the letters in order traces the boundary, and it is how a student tells OAPB from OAQB without looking at the shading.
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Worked answers: Exercise 11.1
Transcript1,390 words
Before any of the words, be clear about what is being cut. Not the curve. The curve is a loop of no thickness, and cutting a loop leaves you with two arcs and nothing else. The thing under discussion is the filled circle: the region, all the way in to the centre. Cutting that leaves you with pieces of area, and it is those pieces that get names. There are two ways to cut it, and each one gets its own word.
The first cut. Mark the centre, pick two points on the rim, and draw the two radii out to them. The piece between those two radii, closed off by the arc that runs from one to the other, is called a sector. Look at what bounds it: two straight edges and one curved one. The two straight edges both run out from the centre, and that is the whole of what makes it a sector rather than anything else.
It carries an angle with it. The two radii open out at the centre, and that opening is the angle of the sector. The second cut. Take the same two points on the rim, and this time join them with a straight line instead. That line is a chord. The piece between the chord and the arc is called a segment. One straight edge now, not two. And notice what has disappeared from the picture: the centre. A segment is described without ever mentioning where the centre is.
Same two points on the rim, same arc. A different way of closing the region off, and a different word. Now the thing both words have in common, and it is easy to walk past. A cut that runs right across a filled circle leaves exactly two pieces. Not one, not three. Two. That is a fact about the cut, not about the word. Draw two radii and the disc falls into two. Draw a chord and it falls into two.
You can check that it is really a fact by trying to break it. A line that misses the circle altogether leaves one piece. A cut that starts at the rim and stops halfway leaves one piece, because you can still walk round the end of it. Two cuts that cross leave four. Three through one point leave six. So two is what one cut right across gives you, and it is worth saying out loud, because everything that follows is bookkeeping on top of it.
Here is the consequence. Neither word names one region. When the two radii cut the disc, both of the pieces they leave are sectors. The shaded one and the unshaded one. Shading is a pointer, not a definition. When the chord cuts it, both of the pieces are segments. So a sector always comes with a partner sector, and a segment always comes with a partner segment. Each word names a pair.
Which means the words are not finished. Given a cut, you still need a way to say which of the two you meant. That is the job the other two words do. The larger of the two is the major one and the smaller is the minor one. Larger by area. Not by how it looks, not by which one was drawn first, and not by the number you happened to be handed.
That last one is where the mistakes live. Being told an angle of two hundred and forty degrees does not make its sector the minor one. Two hundred and forty is more than half a turn, so that sector is the major one and the one left over is the minor. And minor does not mean tiny. A sector of a hundred and seventy nine degrees is still the minor one, because its partner is a hundred and eighty one.
Minor means smaller than its partner. That is all it means. For sectors there is an easy way to get the other one's angle. The two angles at the centre make up a full turn between them, so the partner's angle is what is left of three hundred and sixty. Thirty degrees leaves three hundred and thirty. A right angle leaves two hundred and seventy. A hundred and twenty leaves two hundred and forty.
You can see why without any arithmetic. Stand at the centre and sweep all the way round. Two directions are marked, the two radii, and between them the sweep is split into two runs. Every direction is in one run or the other, and none is in both. So the two runs use up the turn exactly once, and that is the subtraction. The same accounting works on the areas, and this is the part worth keeping.
The two pieces cover the whole disc, and they do not overlap. Nothing is counted twice and nothing is missed. So the major piece is the whole disc take away the minor piece. Exactly, not roughly. That is why nobody teaches you a formula for a major sector. There is nothing new to learn: work out the smaller one and subtract. And it is a consequence rather than a rule. It follows from the two pieces tiling the disc, which follows from there being exactly two of them.
Put numbers on it once, to see the subtraction land. A disc of radius four, with pi taken as three point one four. The whole disc is fifty point two four. Cut it with two radii thirty degrees apart. The minor sector is a twelfth of the disc, which is four point one nine to two places. The major sector is the other three hundred and thirty degrees, eleven twelfths of the disc, which is forty six point zero five.
Add them. Four point one nine plus forty six point zero five is fifty point two four, which is the whole disc back again. That agreement is the evidence. The two pieces really do use the disc up. Now put both cuts on one circle, because that is how questions arrive. Two radii out to the same two rim points, and the chord joining them. Three lines, and now three regions to talk about.
The minor sector is the wedge. The minor segment is the thinner piece past the chord. And between them sits the triangle on the centre and the two rim points. The sector is the segment plus that triangle. Which is the honest answer to whether a sector and a segment are the same thing: they differ by a triangle, and by exactly that. Read it the other way and it still works. The major segment is the major sector plus the same triangle.
There is one case the naming has nothing to say about, and it is better to meet it here than in a question. Send the chord through the centre. Now it is a diameter, and the two pieces are two half discs of exactly the same area. Neither is larger. So neither is major and neither is minor, and asking which is which has no answer. The same thing happens with two radii opened out to a straight angle: a hundred and eighty degrees each side, two equal sectors, no larger one.
And that is the one place where the two words coincide. With the cut through the centre, the triangle between the sector and the segment has flattened to nothing, so the wedge and the piece past the chord are the same half disc. One convention, and then we are done. Because the minor piece is the one you usually want, people often let the bare word stand for it. Say sector with nothing in front of it and the minor sector is meant; say segment and the minor segment is meant.
That is a habit of speech, not a fact about circles, and it is worth knowing that it is a habit. If a question says major, it means the other one, and if it says nothing, it means the smaller. So: two words for the cut, two words for the piece. Two radii and an arc make a sector. A chord and an arc make a segment. Either cut leaves exactly two pieces, the larger is the major one, and it is always the whole disc take away the smaller.
Where this fits
Taken from the notes each video was made from, not from the reading order — these are the ideas this one rests on and the ones that later rest on it.
Builds on
- Counting shared points sorts every line into one of three kindsClass 10 · Ch 10, Circles
Comes up again in
- A sector's area as its share of the full turnClass 10 · Ch 11, Areas Related to Circles
- The arc's length by the same share argumentClass 10 · Ch 11, Areas Related to Circles
- A segment as what is left when the triangle is taken awayClass 10 · Ch 11, Areas Related to Circles
Either side of this one
- Why the two from an outside point are always equally longClass 10 · Ch 10, Circles