In this chapter, we calculate areas and perimeters of plane figures involving circles — including sectors, segments, and combinations of circles with polygons. This is a highly practical chapter useful in design, architecture, and engineering.
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Key Formulas
Let r = radius and theta = angle (in degrees) unless stated otherwise.
- Circle:
- Circumference = 2 pi r
- Area = pi r2
- Sector (a "pie slice" of a circle):
- Arc length = (theta / 360) × 2 pi r
- Area of sector = (theta / 360) × pi r2
- Segment:
- Area of minor segment = Area of sector - Area of triangle (formed by the two radii and chord)
- Area of major segment = Area of circle - Area of minor segment
Key formulas
Use pi = 22/7 or 3.14 as specified in the problem.
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Worked Examples
Find the area of a sector of radius 7 cm with central angle 90°.
- Area = (90/360) × (22/7) × 72 = (1/4) × 22 × 7 = 38.5 sq cm
The minute hand of a clock is 14 cm long. Find the area swept in 5 minutes.
- In 5 minutes, angle swept = (5/60) × 360° = 30°
- Area = (30/360) × (22/7) × 142 = (1/12) × 22 × 28 = 616/12 = 51.33 sq cm
Find the area of the minor segment of a circle of radius 14 cm with chord angle 60° at centre.
- Area of sector = (60/360) × pi × 196 = (1/6) × (22/7) × 196 = 102.67 sq cm
- Area of equilateral triangle (since angle = 60°, it is equilateral with side 14): (sqrt3/4) × 142 = 49 sqrt3 = 84.87 sq cm
- Area of segment = 102.67 - 84.87 = 17.8 sq cm
A horse is tied to a peg at one corner of a square field of side 15 m with a rope of length 5 m. Find the area the horse can graze.
- The horse can graze a quarter circle of radius 5 m.
- Area = (1/4) × pi × 52 = (1/4) × (22/7) × 25 = 19.64 sq m
Find the area of the shaded region if a circle is inscribed in a square of side 14 cm.
- Radius of inscribed circle = 14/2 = 7 cm
- Area of square = 142 = 196 sq cm
- Area of circle = (22/7) × 49 = 154 sq cm
- Shaded area (corners) = 196 - 154 = 42 sq cm
Two circles touch internally. Their radii are 8 cm and 3 cm. Find the area between them.
- Area = pi (82 - 32) = pi (64 - 9) = 55 pi = 55 × (22/7) = 172.86 sq cm
A sector of a circle of radius 12 cm has perimeter 25 cm. Find the area of the sector.
- Perimeter of sector = 2r + arc length => 25 = 24 + arc length => arc length = 1 cm
- Area = (1/2) × r × l = (1/2) × 12 × 1 = 6 sq cm (using the radian formula)
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Common mistakes
> Common mistakes: Students confuse the area of a sector with the area of a segment. The segment does NOT include the triangular part — you must subtract the triangle from the sector. Always check whether the problem uses pi = 22/7 or 3.14. Also, for the horse-grazing type problem, the fraction of the circle depends on the interior angle of the corner.
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Summary
The key formulas are: sector area = (theta/360) × pi r2, arc length = (theta/360) × 2 pi r, and segment area = sector area - triangle area. Most real-life problems combine a circle with a rectangle or square, and require finding areas by addition or subtraction. Always draw a diagram to identify which region is asked for.