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Class 10 Areas Related to Circles: cheat sheet
CBSE Mathematics. Parts of a circle: the length of an arc, the area of a sector and of a segment, and problems with rings, wheels, clock hands and wipers.
Key ideas
- A sector is the region between two radii and their arc; a segment is the region between a chord and its arc.
- An arc or sector of angle θ is θ/360 of the whole circle.
- Minor segment = sector − triangle; major segment = circle − minor segment.
- The area of a ring between circles of radii R and r is π(R² − r²).
- In one turn a wheel moves forward by its circumference, 2πr.
- The minute hand turns 6° every minute; the hour hand 30° every hour.
- Use the value of π the question gives (22/7 suits radii that are multiples of 7).
Remember
| Length of arc | (θ/360) × 2πr |
| Area of sector | (θ/360) × πr² |
| Area of minor segment | (θ/360) × πr² − area of the triangle |
| Triangle cut off by a 90° chord | area = r²/2 |
Watch out
- Using πr² for the arc length or 2πr for the sector area.
- Adding the triangle instead of subtracting it for the minor segment.
- Forgetting the two radii when finding the perimeter of a sector.
Worked example
A wheel has a diameter of 28 cm. How far does it travel in 650 complete turns (π = 22/7)? Give the answer in metres.
Working: one turn = 22/7 × 28 = 88 cm; 650 turns = 57,200 cm
Answer: 572 m
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