Areas Related to Circles

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

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