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§11Class 10, Chapter 11

Areas Related to Circles: Pizza Slices, Pie Charts, and Everything In Between

Every rotating or fanned-out shape — clock hands, umbrella ribs, wipers, lighthouse beams — boils down to a sector or segment of a circle. This companion covers the arc-length and area formulas, the sector-minus-triangle trick for segments, and a set of classic worked problems, with tools, flashcards and a CBSE-style quiz to test yourself.

sector = (θ/360)×πr²arc = (θ/360)×2πrsegment = sector − triangleminor & major

🥧11.1 Sectors and Segments: Naming the Regions

You've already met the words sector and segment of a circle in earlier classes. A sector is the region enclosed by two radii and the arc between them — picture a single slice of a pizza. A segment is the region enclosed by a chord and the arc it cuts off — picture what's left after you slice a flat edge off a circular piece of dough.

minormajor
Minor Sector OAPB
Major Sector OAQB
minormajor
Minor Segment APB
Major Segment AQB

In a circle with centre O, if OAPB is the region enclosed between the two radii OA, OB and the (shorter) arc APB, it's called the minor sector, and ∠AOB is called the angle of the sector. The rest of the circle, OAQB, is the major sector — and its angle is always 360° − ∠AOB, since the two sectors together make up the whole circle.

Similarly, for a chord AB of the circle, the smaller piece cut off (APB) is the minor segment, and the larger remaining piece (AQB) is the major segment.

📏A naming convention worth remembering.. Whenever this chapter (or a question) just says "segment" or "sector" without specifying minor or major, it always means the MINOR one, unless stated otherwise.
🍕
Sector vs. segment, in one picture.. A sector always includes the centre (it's bounded by two straight radii meeting at O). A segment never includes the centre unless the chord happens to be a diameter — it's bounded by a straight chord that cuts across the circle, missing the centre entirely in general.

📐11.1 Formulas for Arc Length, Sector Area, and Segment Area

You already know the area of a full circle is πr². Think of the full circle as one giant sector with a 360° angle at the centre — then use the unitary method to scale down to any smaller angle θ.

START FROM THE FULL CIRCLEWhen the angle at the centre is 360°, the sector's area is πr² (the whole circle).
SCALE DOWN TO 1°When the angle is just 1°, the area is (πr²)/360 — one 360th of the whole circle.
SCALE UP TO θ°When the angle is θ°, the area is θ times as much: (θ/360) × πr².
🥧Area of a Sector. Area of the sector of angle θ = (θ/360) × πr², where r is the radius and θ is the sector's angle in degrees.

The exact same unitary-method reasoning applies to the arc length: scale the full circumference 2πr down proportionally by θ/360.

🌙Length of an Arc. Length of an arc of a sector of angle θ = (θ/360) × 2πr.

Now for the segment. A segment is what's left of a sector once you cut away the straight-edged triangle formed by the two radii and the chord — so its area is just a subtraction.

🌗Area of a Segment. Area of the segment = Area of the sector − Area of the triangle formed by the two radii and the chord = (θ/360) × πr² − area of △OAB.
🔄Getting from minor to major is always a subtraction from the whole circle.. Area of major sector = πr² − area of minor sector. Area of major segment = πr² − area of minor segment. Both follow simply because the minor and major pieces together always make up the complete circle.

For the triangle's area in the segment formula, the most useful general tool is one you've already met: if the chord subtends angle θ at the centre, then △OAB is isosceles (OA = OB = r), so its area works out to ½r²sin θ. For "nice" angles like 60°, 90°, and 120°, this is often easiest to compute directly using the special trigonometric values from Chapter 8.

✏️11.1 Worked Examples

Find the area of the sector of a circle with radius 4 cm and angle 30°. Also find the area of the corresponding major sector. (π = 3.14)

MINOR SECTORArea = (30/360) × 3.14 × 4² = (1/12) × 3.14 × 16 = 12.56/3 ≈ 4.19 cm²
MAJOR SECTOR — METHOD 1Area = πr² − minor sector area = (3.14×16) − 4.19 = 50.24 − 4.19 = 46.05 ≈ 46.1 cm²
MAJOR SECTOR — METHOD 2 (CROSS-CHECK)Area = [(360−30)/360] × 3.14 × 16 = (330/360) × 50.24 ≈ 46.1 cm² ✓ — both methods agree.

📝11.2 Chapter Summary

  1. Length of an arc of a sector with radius r and angle θ (in degrees): (θ/360) × 2πr.
  2. Area of a sector with radius r and angle θ (in degrees): (θ/360) × πr².
  3. Area of a segment: Area of the corresponding sector − Area of the corresponding triangle.
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These formulas are everywhere once you notice them.. Clock hands sweeping out sectors, umbrella ribs, car wipers, lighthouse beams, pizza slices, pie charts — any time something rotates or fans out from a fixed point, sector and segment area formulas are quietly doing the work behind the scenes.