Circumference and area of a circle (review)
For a circle of radius r, the circumference is 2πr and the area is πr². π is about 22/7 or 3.14; use the value the question gives.
Formulas for circles
| Figure | Perimeter | Area |
|---|---|---|
| Circle of radius r | 2πr | πr² |
| Semicircle | πr + 2r (curved edge + diameter) | πr²/2 |
| Quadrant (quarter circle) | πr/2 + 2r | πr²/4 |
| Ring between radii R and r | 2πR + 2πr (both edges) | π(R² − r²) |
Choosing π and units
Your textbook now starts this chapter directly with sectors and segments. So you will mostly use these formulas inside those questions, for example to find r from a given circumference. Use π = 22/7 when the radius or diameter is a multiple of 7. Use π = 3.14 when the question says so. If no value is given, use 22/7 and write that you did. A length (circumference, perimeter) is in cm or m. An area is in cm² or m². Round decimal answers to 2 decimal places unless told otherwise.
Circumference and area
Find the circumference and the area of a circle of radius 21 cm. (Use π = 22/7)
- Circumference = 2πr = 2 × (22/7) × 21.
- = 2 × 22 × 3 = 132 cm.
- Area = πr² = (22/7) × 21 × 21.
- = 22 × 3 × 21 = 1386 cm².
Circumference = 132 cm, area = 1386 cm²
Word problem: bicycle wheel
A bicycle wheel has diameter 70 cm. How many complete revolutions does it make to travel 1.1 km? (Use π = 22/7)
- Distance in one revolution = circumference = πd = (22/7) × 70 = 220 cm.
- 1.1 km = 1100 m = 110000 cm.
- Number of revolutions = 110000 ÷ 220 = 500.
500 revolutions
Word problem: a path round a park
A circular park has radius 35 m. A path 7 m wide runs all around it, outside. Find the area of the path. (Use π = 22/7)
- Inner radius r = 35 m; outer radius R = 35 + 7 = 42 m.
- Area of the ring = π(R² − r²).
- R² − r² = 1764 − 1225 = 539.
- Area = (22/7) × 539 = 22 × 77 = 1694 m².
1694 m²
A roti of radius 14 cm covers (22/7) × 196 = 616 cm² of the tawa. An auto-rickshaw meter finds distance from wheel turns: distance = turns × circumference.
1 marks: “Find the area of a circle whose circumference is 44 cm. (Use π = 22/7)” 2πr = 44 gives r = 7 cm. Area = (22/7) × 49 = 154 cm².
2 marks: “The radii of two circles are 15 cm and 13 cm. Find the radius of the circle whose circumference is the sum of their circumferences.” 2πR = 2π(15) + 2π(13), so R = 15 + 13 = 28 cm.
3 marks: “A wheel has diameter 56 cm. How many revolutions will it make to cover 352 m? (Use π = 22/7)” Circumference = (22/7) × 56 = 176 cm. 352 m = 35200 cm. Revolutions = 35200 ÷ 176 = 200.
Circumference wraps around (2πr, in cm). Area fills inside (πr², in cm²).