CBSE Class 10 · Mathematics
Areas Related to Circles
Official NCERT chapter from Mathematics (book code jemh1). ExamMaster notes are original teaching at CBSE Class 10 depth.
This lesson follows the official chapter “Areas Related to Circles” in Mathematics. The words below are ExamMaster’s teaching, not a paste from the book. Use the NCERT chapter for the classroom sequence; use these notes to hold the idea without copying exercises or figures.
- CBSE Class 10
- Medium level
- 8 concepts
1Areas of Sector and Segment of a Circle
A sector is a pie-slice: (θ/360) of the disc’s area, πr². A segment is the bite between a chord and its arc: sector minus (or plus) the triangle, as taught. θ is the centre-angle. A half-disc is θ = 180°.
Using the chord-length as if it were the sector is a miss.
Figure. A 70° sector is the 70/360 slice of the disk: area (70/360)πr². The segment is that sector minus the triangular bite with the two radii and the chord. Bar lengths are the angles, not the areas.
How it works
- Name r and the centre-angle θThe given.
- Sector = (θ/360)πr²The slice.
- Segment = sector ± triangle as taughtThe bite.
Quarter of a 14 cm circle
Using π = 22/7, area of a 90° sector of radius 14 cm.
- Fraction90/360 = 1/4
- πr²(22/7)×196 = 616
- Sector154 cm²
Pro tip. 22/7 is a stand-in; the sector is the fraction of the disc.
2Mark what is given on the figure
Mark what is given on the figure: r, θ, a chord, a triangle’s height. A missing θ is not 90° because the slice “looked like a quarter”. Marks are the given; a look is not a mark.
Inventing 60° from a freehand pie is a silent given.
Figure. Mark the given first: the radius to the contact point is perpendicular to the tangent. The two lines are axis-aligned, so the right angle is a true 90° on this canvas. The circle outline is not drawn.
How it works
- Copy r and θ (or the chord) onto the sketchThe marks.
- Name sector or segment as the unknownThe job.
- Refuse a look-only angleGiven is marked.
3A definition is a test, not a picture
Sector / segment is a test: a named centre-angle plus r, or a chord plus the slice-rule. If you only shade a “bite”, you have a doodle.
A green crayon region is not the test.
Figure. A definition is a test you run on a point: inside the disk you can draw no tangent, on the circle exactly one, outside exactly two. The test is the count, not a pretty outline.
How it works
- Name θ and r, or the chord and the ruleThe object.
- Write the fraction of πr² or the sector-minus-triangleThe test.
- Then the word has contentThe definition ran.
4Equal marks must be earned
Equal marks must be earned: equal radii, equal θ, or a taught isosceles from two radii. A tick you did not justify is not equal. Two radii of the same circle are equal — that one you may use.
Ticking a chord equal to a radius because it “looked it” is unearned.
Figure. Two tangents from one external point are equal in length — the marks are earned by that theorem, not by the drawing looking tidy. Both segments are the same visual length (aspect 1.6). The circle is omitted.
How it works
- Name which marks you claim equalThe claim.
- Give the reason (same circle; given; taught)Earned.
- Drop a tick that has no reasonHonest marks.
5Perimeter is the walk around
Perimeter here can mean the walk of a sector’s boundary: two radii plus the arc (θ/360)×2πr. It is not the area. A segment’s walk, if asked, is chord plus arc — not two radii unless that was the sector.
Adding πr² into a perimeter is the cover-steal.
Figure. Perimeter is the walk around the edge: 6+4+6+4=20 of the same unit. The rectangle is drawn 3:2 (visual sides 6 and 4). Area tiles are a different count.
How it works
- Name which edges are walkedRadii + arc, or chord + arc.
- Add those lengthsThe walk.
- Keep cm, not cm²Perimeter.
6Area is the tiles inside
Area is the tiles inside: the sector’s fraction of πr², or the segment’s leftover. A 90° sector of r = 14 (π = 22/7) is 154 cm² — the cover, not the walk 22+14+14.
Using 2πr when area was asked is the walk-steal.
Figure. Area counts the unit tiles that fill the inside. Six visually square tiles (side corrected for aspect 1.8). The walk around the edge is a different number.
How it works
- Write the fraction of the disc or the sector-minus-triangleThe cover.
- Compute with a named πThe tiles.
- Keep cm²Area.
7A formula names a count of units
A formula names a count of units: (θ/360)πr² is a count of unit squares in the slice. 22/7 is a stand-in for π, labelled as such. The formula is not a decoration around a shaded pie.
Treating 22/7 as exactly π is a fake close.
Figure. A formula names a count of unit squares: πr² with r=7 is 49π square units. It is not a decoration beside a picture. Keep π unless a numerical π is asked.
How it works
- Write the formula and the unitcm² or cm.
- Name π as approximate if you use 22/7The stand-in.
- Refuse a decoration that skips the countA named count.
8Parallel lines never meet and stay equally far
Parallel lines never meet and stay equally far — use that only if a taught figure put a chord parallel to something, or if a tangent-chord story needs it. This chapter’s main job is still the slice. A parallel mark you did not earn is a steal from a different heading.
Declaring two radii parallel is usually a miss — they meet at O.
Figure. Parallel lines stay a constant distance: the two vertical brackets are the same span d at two places, so the gap does not change. They do not meet on the page.
How it works
- See whether a parallel was actually markedThe given.
- If yes, use never-meet / equal-distance as taughtThe parallel.
- If no, return to sector and segmentThis chapter’s job.
A 90° sector of radius 14 cm (π = 22/7) has area
- 154 cm²
- 44 cm
- 616 cm²
A quarter of 616; 44 cm would be a walk.
Notes
- The official chapter title is “Areas Related to Circles”. Teach the school test for that title, not a contest shortcut.
- If a step needs a later class, stop. The next official chapter will pick it up.
Formulas
- sector A = (θ/360)πr²
- arc = (θ/360)2πr
- segment = sector ± triangle (as taught)
Recap
Hold these pegs from the official chapter “Areas Related to Circles”. The wording is ExamMaster’s teaching, not a textbook recap.
- Areas of Sector and Segment of a Circle
- A sector is a pie-slice: (θ/360) of the disc’s area, πr².
- Mark what is given on the figure
- Mark what is given on the figure: r, θ, a chord, a triangle’s height.
- A definition is a test, not a picture
- Sector / segment is a test: a named centre-angle plus r, or a chord plus the slice-rule.
- Equal marks must be earned
- Equal marks must be earned: equal radii, equal θ, or a taught isosceles from two radii.
- Perimeter is the walk around
- Perimeter here can mean the walk of a sector’s boundary: two radii plus the arc (θ/360)×2πr.
- Area is the tiles inside
- Area is the tiles inside: the sector’s fraction of πr², or the segment’s leftover.
Practise Areas Related to Circles
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