What this chapter is about
This chapter extends your knowledge of circles from perimeter and area of a full circle to finding areas and perimeters of parts of circles. You learn to calculate the area of a sector (a slice of a circle, like a pizza slice) and the area of a segment (the region between a chord and its arc). These concepts connect your understanding of circles with central angles measured in degrees.
The chapter also teaches you to find areas of combined figures — shapes made by joining or removing circular parts from rectangles, triangles or other circles. Such calculations appear in real-life problems: designing circular gardens, finding material needed for a wheel's rubber rim, or calculating the area of a table mat with curved edges.
After studying this chapter, you should be able to find arc lengths, sector areas and segment areas when given the radius and central angle. You should also handle composite figures where you add or subtract areas of different shapes.
Key ideas
- The circumference of a circle with radius r is 2πr, and its area is πr².
- An arc is a part of the circumference; a sector is the region enclosed by two radii and the arc between them.
- The length of an arc subtending angle θ at the centre is (θ/360) × 2πr.
- The area of a sector with central angle θ is (θ/360) × πr².
- A chord divides a circle into two segments; the minor segment is the smaller region, the major segment is the larger.
- Area of a segment = Area of the corresponding sector − Area of the triangle formed by the two radii and the chord.
- For combined figures, identify which areas to add and which to subtract, then compute each part separately.
- Always use consistent units and substitute π = 22/7 or 3.14 as directed in the problem.
Formulas and facts to remember
- Quantity: Circumference of circle · Formula: 2πr · Meaning: Total length around the circle
- Quantity: Area of circle · Formula: πr² · Meaning: Region enclosed by the circle
- Quantity: Length of arc · Formula: (θ/360) × 2πr · Meaning: Part of circumference for angle θ degrees
- Quantity: Area of sector · Formula: (θ/360) × πr² · Meaning: Slice of circle for angle θ degrees
- Quantity: Area of segment · Formula: Area of sector − Area of triangle · Meaning: Region between a chord and its arc
- Quantity: Area of major sector · Formula: πr² − Area of minor sector · Meaning: Remaining part of circle
- Quantity: Perimeter of sector · Formula: 2r + arc length · Meaning: Two radii plus the curved part
Worked examples
Example 1: A circular park has radius 21 metres. A path runs along an arc that subtends an angle of 60° at the centre. Find the length of this path. (Use π = 22/7)
Step 1: Identify given values. Radius r = 21 m, central angle θ = 60°.
Step 2: Use the arc length formula. Arc length = (θ/360) × 2πr = (60/360) × 2 × (22/7) × 21
Step 3: Simplify. = (1/6) × 2 × (22/7) × 21 = (1/6) × 2 × 22 × 3 = (1/6) × 132 = 22 m
The path is 22 metres long.
Example 2: A sector of a circle has radius 14 cm and central angle 90°. Find the area of this sector. (Use π = 22/7)
Step 1: Write the formula. Area of sector = (θ/360) × πr²
Step 2: Substitute values. = (90/360) × (22/7) × 14 × 14 = (1/4) × (22/7) × 196
Step 3: Calculate. = (1/4) × 22 × 28 = (1/4) × 616 = 154 cm²
The sector's area is 154 cm².
Example 3: A chord of a circle of radius 7 cm subtends an angle of 60° at the centre. Find the area of the minor segment. (Use π = 22/7 and √3 = 1.73)
Step 1: Find area of sector. Area of sector = (60/360) × (22/7) × 7² = (1/6) × (22/7) × 49 = (1/6) × 154 = 25.67 cm² (approx.)
Step 2: Find area of the triangle formed by the two radii and the chord. The triangle has two sides equal to 7 cm with included angle 60°. Area of triangle = (1/2) × r × r × sin 60° = (1/2) × 7 × 7 × (√3/2) = (49 × 1.73)/4 = 21.22 cm² (approx.)
Step 3: Area of segment = Area of sector − Area of triangle = 25.67 − 21.22 = 4.45 cm² (approx.)
The minor segment's area is approximately 4.45 cm².
Common mistakes
Confusing arc length with sector area → Arc length is a distance (linear), sector area is a region (square units); use the correct formula for what is asked.
Forgetting to subtract the triangle's area when finding a segment → A segment is smaller than the sector; always subtract the triangular part.
Using diameter instead of radius in formulas → Check whether the problem gives radius or diameter; if diameter is given, divide by 2 before substituting.
Mixing up minor and major sectors → Minor sector has the smaller angle (≤180°); major sector has the larger angle; their areas add up to πr².
Not converting angle to degrees when needed → These formulas assume angle in degrees; if angle is in some other form, convert first.
Quick revision
- Arc length = (θ/360) × 2πr; sector area = (θ/360) × πr².
- Segment area = Sector area − Triangle area formed by the radii and chord.
- For combined figures, break into familiar shapes, find each area, then add or subtract.
- Major sector area + Minor sector area = Full circle area.
- Always check units and whether radius or diameter is given.