Learning Objectives
- Find the perimeter and area of a circle
- Find the area of a sector and the length of an arc
- Find the area of a segment of a circle
- Solve problems involving combinations of plane figures
Key Concepts
Circle Formulas
- Circumference = 2πr
- Area of circle = πr²
- Area of semi-circle = πr²/2
- Area of quadrant = πr²/4
Sector of a Circle
A sector is the region enclosed between two radii and the corresponding arc.
- Length of arc = (θ/360°) × 2πr
- Area of sector = (θ/360°) × πr²
where θ is the angle of the sector at the centre in degrees.
Segment of a Circle
A segment is the region between a chord and its corresponding arc.
Area of minor segment = Area of sector - Area of triangle
Area of major segment = Area of circle - Area of minor segment
For a sector with angle θ and radius r:
Area of segment = (θ/360°) × πr² - ½r² sin θ
Combinations of Plane Figures
Many problems involve finding areas of shaded regions by adding or subtracting areas of circles, sectors, triangles, rectangles, and other standard shapes.
Summary
Areas related to circles extend the basic area formula to sectors (pie-shaped regions) and segments (chord-bounded regions). The angle at the centre determines what fraction of the full circle is involved. Problems often require combining areas of different shapes through addition or subtraction.
Important Terms
- Sector
- The region enclosed between two radii and the corresponding arc of a circle
- Segment
- The region between a chord and its corresponding arc
- Minor Arc
- The shorter arc between two points on a circle
- Major Arc
- The longer arc between two points on a circle
Quick Revision
- Area of sector = (θ/360) × πr²
- Length of arc = (θ/360) × 2πr
- Area of segment = Area of sector - Area of triangle formed
- Area of major segment = πr² - Area of minor segment
- Use π = 22/7 or 3.14 as specified in the problem