Parts of a circle you need
- Arc: a piece of the circle's edge (like pizza crust).
- Chord: a straight line joining two points on the circle.
- Sector: the region between two radii and an arc. It looks like a pizza slice. The angle at the centre is θ.
- Segment: the region between a chord and an arc. It looks like the crust strip left after cutting straight across.
- Minor means the smaller piece; major means the bigger piece.
Remember: area of a full circle = πr², circumference = 2πr.
Area of a sector
A full turn is 360°. A sector with angle θ is θ out of 360 of the circle. So it gets the same share of the area.
Area of sector = θ/360 × πr²
- θ = 90° → 1/4 of the circle (a quadrant)
- θ = 60° → 1/6 of the circle
- θ = 120° → 1/3 of the circle
Major sector = πr² − minor sector. Its angle is 360° − θ.
Length of an arc
The arc gets the same share of the edge: arc length = θ/360 × 2πr.
Perimeter of a sector = arc + 2 radii = θ/360 × 2πr + 2r. Many students forget the two radii.
Area of a segment
The segment is what is left when you cut the triangle OAB out of the sector.
Minor segment = sector − triangle OAB
Major segment = πr² − minor segment
Triangle area for the three syllabus angles
- θ = 90°: the triangle is right-angled. Area = ½ × r × r = r²/2.
- θ = 60°: OA = OB = r and the angle is 60°, so the triangle is equilateral. Area = (√3/4) r².
- θ = 120°: draw the height from O to AB. It is r/2 (half of r) and AB = √3 r. Area = ½ × √3 r × r/2 = (√3/4) r².
General rule (for checking): triangle = ½ r² sin θ.
Try it at home
Draw a circle of radius 7 cm on squared paper (1 cm squares). Cut out a 90° slice. Count the full squares plus about half of the broken ones. You will get close to 38.5. Now fold the slice along the chord: the part that sticks out past the fold is the segment, about 14 squares (38.5 − 24.5).
Key formulas and definitions
- Area of circle = πr², circumference = 2πr
- Area of sector = θ/360 × πr²
- Arc length = θ/360 × 2πr
- Perimeter of sector = arc length + 2r
- Minor segment = sector − triangle
- Major segment = πr² − minor segment; major sector = πr² − minor sector
- Triangle: 90° → r²/2; 60° or 120° → (√3/4) r²
Worked examples
1. Find the area of a circle of radius 7 cm. (π = 22/7)
A = πr² = 22/7 × 7 × 7 = 22 × 7 = 154 cm².
2. Find the area of a quadrant (90° sector) of a circle of radius 14 cm.
Whole circle = 22/7 × 14 × 14 = 616 cm². Quadrant = 90/360 × 616 = 1/4 × 616 = 154 cm².
3. Find the arc length of a 60° sector of radius 21 cm.
Circumference = 2 × 22/7 × 21 = 132 cm. Arc = 60/360 × 132 = 1/6 × 132 = 22 cm.
4. A clock's minute hand is 14 cm long. What area does it sweep in 5 minutes?
60 minutes = 360°, so 5 minutes = 30°. Area = 30/360 × 22/7 × 14 × 14 = 1/12 × 616 = 51.33 cm² (about).
5. A chord subtends 90° at the centre of a circle of radius 10 cm. Find the minor and major segments. (π = 3.14)
Sector = 1/4 × 3.14 × 100 = 78.5 cm². Triangle = ½ × 10 × 10 = 50 cm². Minor segment = 78.5 − 50 = 28.5 cm². Circle = 314 cm², so major segment = 314 − 28.5 = 285.5 cm².
6. Radius 12 cm, angle 60°. Find the minor segment. (π = 3.14, √3 = 1.73)
Sector = 1/6 × 3.14 × 144 = 75.36 cm². The triangle is equilateral: √3/4 × 144 = 1.73 × 36 = 62.28 cm². Segment = 75.36 − 62.28 = 13.08 cm².
7. Radius 21 cm, angle 120°. Find the minor segment. (π = 22/7, √3 = 1.73)
Sector = 1/3 × 22/7 × 441 = 462 cm². Triangle = √3/4 × 441 = 1.73 × 110.25 = 190.73 cm². Segment = 462 − 190.73 = 271.27 cm².
Common mistakes
- Forgetting to divide by 360. θ/360 is the share of the circle, not θ itself.
- Using 2πr (edge) for area or πr² for arc. Area has r², length has r.
- Giving the sector as the segment. Segment = sector − triangle; do not stop halfway.
- Leaving out the two radii when asked for the perimeter of a sector.