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Areas Related to Circles is one of the shorter Class 10 chapters, and most of its marks come from two formulas used carefully. This post gives you the formulas, three fully worked examples at the angles CBSE allows, the mistakes that lose marks, and a short practice plan.
What does the chapter cover now?
The current NCERT chapter (2026-27 reprint) has one main section: Areas of Sector and Segment of a Circle, followed by a summary and one exercise. Area of combinations of plane figures is not part of it. CBSE's Class 10 Mathematics curriculum lists "Area of sectors and segments of a circle" and adds a clear limit: "In calculating area of segment of a circle, problems should be restricted to central angle of 60°, 90° and 120° only." Check the curriculum for your exam year, because CBSE issues one every session.
What are the formulas?
From the NCERT summary, for a circle of radius r and a sector of angle θ degrees:
Length of arc = (θ/360) × 2πr
Area of sector = (θ/360) × πr²
Area of segment = area of sector − area of the triangle formed by the two radii and the chord
A sector is the "pizza slice" between two radii. A segment is the region between a chord and its arc. The minor segment goes with the smaller arc; the major segment is the rest of the circle.
Worked example 1: a 60° sector and segment
A circle has radius 14 cm and a chord subtends 60° at the centre. Use π = 22/7 and √3 = 1.73.
Arc length = (60/360) × 2 × (22/7) × 14 = (1/6) × 88 = 14.67 cm.
Sector area = (1/6) × (22/7) × 196 = (1/6) × 616 = 102.67 cm².
With a 60° angle and two equal radii, the triangle is equilateral, side 14 cm. Area = (√3/4) × 14² = 49√3 = 49 × 1.73 = 84.77 cm².
Minor segment = 102.67 − 84.77 = 17.90 cm².
Worked example 2: a 90° segment and the major segment
A circle of radius 7 cm has a chord subtending 90° at the centre. Use π = 22/7.
Sector (a quadrant) = (1/4) × (22/7) × 49 = 38.5 cm².
The triangle is right-angled with legs 7 and 7: area = (1/2) × 7 × 7 = 24.5 cm².
Minor segment = 38.5 − 24.5 = 14 cm².
Whole circle = (22/7) × 49 = 154 cm², so major segment = 154 − 14 = 140 cm².
Worked example 3: a 120° segment, using the altitude
Radius 10 cm, central angle 120°. Use π = 3.14 and √3 = 1.73.
Sector = (120/360) × 3.14 × 100 = 104.67 cm².
For the triangle, drop a perpendicular from the centre O to the chord AB, meeting it at M. It bisects the 120° angle, so ∠AOM = 60°.
OM = r cos 60° = 10 × 1/2 = 5 cm, and AM = r sin 60° = 10 × √3/2 = 5√3 cm, so AB = 10√3 cm.
Triangle area = (1/2) × AB × OM = (1/2) × 10√3 × 5 = 25√3 = 25 × 1.73 = 43.25 cm².
Segment = 104.67 − 43.25 = 61.42 cm².
The altitude method works for all three allowed angles, so learn it once.
How do you work backwards from an arc length?
Some questions give the arc and ask for the angle or the sector area. Rearrange the arc formula: θ = (arc length × 360) ÷ (2πr).
Worked example 4. An arc of length 22 cm lies on a circle of radius 21 cm. Find the angle and the sector area. Use π = 22/7.
Circumference = 2 × (22/7) × 21 = 132 cm.
θ = 22 × 360 ÷ 132 = 60°.
Sector area = (60/360) × (22/7) × 441 = (1/6) × 1386 = 231 cm².
A shortcut you can check: sector area = (1/2) × arc length × r = (1/2) × 22 × 21 = 231 cm². Both routes agree, which is a good way to catch slips. For the trigonometric ratios it uses, see our introduction to trigonometry for Class 10.
What mistakes cost marks?
Wrong angle for clock problems. The minute hand turns 360° in 60 minutes, so 6° per minute. In 10 minutes a 7 cm hand sweeps 60°: area = (1/6) × (22/7) × 49 = 25.67 cm².
Diameter used as radius. Read the question twice and halve the diameter first.
Using the wrong π. The NCERT exercise says to use 22/7 unless told otherwise; some questions ask for 3.14. Use what the question asks.
Arc length and area mixed up. Arc uses 2πr and gives cm; area uses πr² and gives cm².
Rounding too early. Keep fractions until the last step, then round.
A one-week practice plan
Day 1: write the three formulas from memory, then the altitude method for 60°, 90° and 120°.
Days 2 and 3: all sector and arc questions in the NCERT exercise.
Days 4 and 5: segment questions at each of the three angles, including one major segment.
Day 6: mixed set including clock and grazing-animal (rope) problems.
Day 7: timed set of five questions, then check units and π on every answer.
For more help with circle geometry, read our post on circles and tangents in Class 10. If you keep setting up segment problems wrongly, the Fermi Problem Library has worked problems with a step-by-step solver that explains the reasoning, so you can see which triangle to subtract and why.
Tonight, solve the 120° example yourself with radius 21 cm and π = 22/7, then check that your segment is smaller than your sector.
