Areas Related to Circles | Maths Grade 10 CBSE

Play with it

Sweep a sector

Drag the angle θ. The shaded sector grows, and its area = (θ/360)·πr², the arc length = (θ/360)·2πr, and the segment (sector minus triangle) all update.

sector · arc · segment (r = 7)drag the angle

angle θ 90°

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The three ideas in this chapter

Two formulas you already know carry this whole chapter:

Area = π r² Circumference = 2 π r

Everything else — sectors, arcs, segments — is just a fraction of these.

Which π?

Use π = 22/7 when the radius is a multiple of 7 (the numbers come out clean), and π ≈ 3.14 otherwise. The question usually tells you which.

Common mistake: swapping the formulas. Area uses r² (πr²); circumference is linear (2πr). If you see an r², it's an area.

A sector is the "pizza slice" between two radii and the arc. Since a full circle is 360°, a sector of angle θ is the fraction θ/360 of the whole:

Sector area = (θ/360) · π r² Arc length = (θ/360) · 2 π r

Worked example · sector area & arc

Find the area and arc length of a 90° sector of a circle of radius 7 cm (take π = 22/7).

Fraction = 90/360 = 1/4.
Area = (1/4)·(22/7)·7² = (1/4)·(22/7)·49 = (1/4)·154 = 38.5 cm².
Arc = (1/4)·2·(22/7)·7 = (1/4)·44 = 11 cm.

A segment is the region between a chord and its arc. To find it, take the sector and cut away the triangle formed by the two radii and the chord:

Segment area = (sector area) − (area of triangle)

The triangle's area (two radii r with angle θ between them) is ½ r² sin θ.

Worked example · minor segment

Find the area of the minor segment cut by a 90° sector of a circle of radius 10 cm (π ≈ 3.14).

Sector area = (90/360)·3.14·100 = (1/4)·314 = 78.5 cm².
Triangle area = ½·10²·sin 90° = ½·100·1 = 50 cm².
Segment = 78.5 − 50 = 28.5 cm².

Common mistake: forgetting to subtract the triangle. A segment is not a sector — it's the leftover after the triangle is removed.

Why this matters

Where you'll actually use this

Circles are everywhere round things move or get shared — and "how much area?" is a question that pays. Pizza, wipers, pie charts and sprinklers all live in this chapter.

Wipers, fans & sprinklers

A windscreen wiper of length r sweeping through an angle θ clears a sector of area (θ/360)·πr². Designers choose the blade length and sweep angle to clear the most glass. Lawn sprinklers and ceiling fans are sized the same way.

Sector area

Pie charts & sharing

Every slice of a pie chart is a sector whose angle is that category's share of 360°. Splitting a pizza fairly, dividing land, or showing a budget breakdown all use the same θ/360 idea you just learned.

θ ∝ share of 360°

🕐 Clock & gauges

A clock hand or speedometer needle sweeps a sector; its area and arc come straight from θ/360.

🏟️ Stadium & tracks

Curved running-track ends are semicircles; designers compute their arc lengths for fair lanes.

🍩 Rings & washers

The area of a ring (annulus) is the big circle minus the small one — a direct πr² subtraction.

🔒 More real-world applications

Clocks, running tracks, rings and more — each explained with a diagram. Free to unlock.

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Check yourself

Competency quiz

Modelled on CBSE's competency pattern — MCQ, assertion–reason and case-study items.

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