Areas Related to Circles
This chapter is about measuring parts of a circle. Starting from the area (πr²) and circumference (2πr), you learn to find the length of an arc, the area of a sector (a 'pizza slice'), and the area of a segment. The formulas all scale the whole-circle value by the fraction θ/360, which makes them easy to remember.
Learning objectives
- Recall the area and circumference of a circle.
- Find the length of an arc for a given central angle.
- Find the area of a sector of a circle.
- Find the area of a segment of a circle.
- Apply the formulas to circular regions and designs.
Key concepts
Area and circumference
For a circle of radius r, the area is πr² and the circumference (perimeter) is 2πr. The value of π is taken as 22/7 or 3.14 as instructed.
Length of an arc
An arc subtending a central angle θ (in degrees) is the fraction θ/360 of the whole circumference: arc length = (θ/360) × 2πr.
Area of a sector
A sector is the region between two radii and the arc between them. Its area is the same fraction of the whole area: area of sector = (θ/360) × πr².
Area of a segment
A segment is the region between a chord and its arc. Area of a (minor) segment = area of the corresponding sector − area of the triangle formed by the two radii and the chord.
Important formulas
Area of circle
πr²
Circumference
2πr
Length of arc
(θ/360) × 2πr
Area of sector
(θ/360) × πr²
Area of segment
area of sector − area of triangle
Key definitions
- Sector
- The region of a circle enclosed by two radii and the arc between them.
- Segment
- The region of a circle enclosed by a chord and the arc it cuts off.
- Arc
- A part of the circumference of a circle.
Solved examples
Q1. Find the area of a circle of radius 7 cm. (Take π = 22/7.)
Solution: Area = πr² = (22/7) × 7 × 7 = 22 × 7 = 154 cm².
Q2. Find the area of a sector of radius 14 cm with central angle 90°. (π = 22/7.)
Solution: Area = (θ/360) × πr² = (90/360) × (22/7) × 14 × 14 = (1/4) × 616 = 154 cm².
Q3. Find the length of an arc of radius 21 cm subtending 60° at the centre. (π = 22/7.)
Solution: Arc = (θ/360) × 2πr = (60/360) × 2 × (22/7) × 21 = (1/6) × 132 = 22 cm.
Common mistakes to avoid
- Mixing up area (πr²) with circumference (2πr).
- Forgetting the θ/360 fraction for arcs and sectors.
- Using the diameter in place of the radius.
- For a segment, forgetting to subtract the triangle's area.
Areas Related to Circles — MCQ Quiz
10 questions with instant feedback. Use number keys 1–4 to answer.
The area of a circle of radius r is:
Practice questions
Short answer
Write the formula for the area of a sector.
(θ/360) × πr², where θ is the central angle.
Find the circumference of a circle of radius 21 cm (π = 22/7).
2 × (22/7) × 21 = 132 cm.
What is a segment of a circle?
The region between a chord and the arc it cuts off.
Long answer
A chord of a circle of radius 10 cm subtends a right angle at the centre. Find the area of the corresponding minor segment. (π = 3.14)
Area of sector = (90/360) × 3.14 × 100 = 78.5 cm². Area of triangle = ½ × 10 × 10 = 50 cm². Segment = 78.5 − 50 = 28.5 cm².
Find the area of a sector of radius 7 cm whose arc length is 11 cm. (π = 22/7)
Arc = (θ/360) × 2πr ⇒ 11 = (θ/360) × 44 ⇒ θ/360 = 1/4. Area = (1/4) × πr² = (1/4) × 154 = 38.5 cm².
HOTS (Higher Order Thinking)
A square of side 14 cm has a circle inscribed in it. Find the area between the square and the circle. (π = 22/7)
Circle radius = 7 cm, area = 154 cm². Square area = 14² = 196 cm². Area between them = 196 − 154 = 42 cm².
Why do the arc, sector and circumference formulas all contain the fraction θ/360?
Because a full circle corresponds to 360°, an angle θ represents the fraction θ/360 of the whole, so each whole-circle quantity is scaled by that fraction.
Quick revision
Revision notes
- Area = πr²; circumference = 2πr.
- Arc length = (θ/360) × 2πr.
- Sector area = (θ/360) × πr².
- Segment area = sector area − triangle area.
Key takeaways
- Everything scales the whole-circle value by θ/360.
- Use the radius, never the diameter, inside the formulas.
- For a segment, always subtract the triangle.
Frequently asked questions
What value of π should I use?
Use 22/7 or 3.14 as stated in the question; many questions are designed to give whole numbers with 22/7.
What is the difference between a sector and a segment?
A sector is bounded by two radii and an arc; a segment is bounded by a chord and an arc.
How do I find the area of a segment?
Find the area of the corresponding sector, then subtract the area of the triangle formed by the two radii and the chord.