Class 10 Mathematics · Chapter 11 NotesAreas Related to Circles

The part of a circular region enclosed by two radii and the corresponding arc is called a sector. In a circle with centre O, the region OAPB bounded by radii OA, OB and arc APB is a sector, and angle AOB is called the angle of the sector. The larger region OAQB is the major sector, and its angle is 360° minus the angle of the minor sector. The part of the circular region enclosed between a chord and the corresponding arc is called a segment. A chord AB divides the circle into a minor segment APB and a major segment AQB. Unless stated otherwise, the words sector and segment mean the minor sector and the minor segment.

• Sectorregion enclosed by two radii and the corresponding arc.
• Angle of a sectorthe angle between its two radii at the centre.
• Segmentregion enclosed between a chord and the corresponding arc.
• Angle of the major sector = 360° − angle of the minor sector.

The boundary of a whole circle is its circumference, 2πr, and it corresponds to an angle of 360° at the centre. If the angle at the centre is reduced to 1°, the arc length becomes 2πr divided by 360. Therefore, for a sector whose angle is θ degrees, the length of the arc is θ/360 × 2πr. This formula is useful whenever you need the distance along a curved edge, for example the distance moved by the tip of a clock hand or the length of wire bent into an arc. Notice that the arc length depends on both the radius and the angle of the sector.

• Length of an arc of sector angle θ = θ/360 × 2πr
• Circumference of the full circle = 2πr, corresponding to 360°.

The area of a complete circular region of radius r is πr², and this may be viewed as a sector of angle 360°. Applying the unitary method, the area of a sector of angle 1° is πr²/360, so the area of a sector of angle θ is θ/360 × πr². This single relation covers every sector: put θ = 90° for a quadrant, θ = 180° for a semicircle, and θ = 360° to recover the whole circle. To find the area of the major sector, either use the angle 360° − θ in the same formula or subtract the area of the minor sector from πr².

• Area of a sector of angle θ = θ/360 × πr²
• Area of major sector = πr² − area of minor sector.
• Area of a quadrant = 1/4 πr²; area of a semicircle = 1/2 πr².

A segment is bounded by a chord and an arc, so its area is found by taking the area of the sector formed by the same arc and removing the triangular region between the two radii and the chord. In symbols, area of segment APB = area of sector OAPB − area of triangle OAB. The triangle OAB is isosceles, since OA and OB are both radii. To find its area, drop a perpendicular from the centre O to the chord AB; this perpendicular bisects the chord and also bisects the angle at the centre. The height and half-chord are then found using the trigonometric ratios of half the sector angle.

• Area of segment = area of the corresponding sector − area of the corresponding triangle.
• Area of major segment = πr² − area of minor segment.
• The perpendicular from the centre to a chord bisects the chord.

For a segment cut off by a chord subtending an angle θ at the centre of a circle of radius r, the triangle formed by the two radii and the chord is isosceles with two sides equal to r and included angle θ. Drawing the perpendicular OM from O to the chord AB gives right triangles OMA and OMB. Here angle AOM = θ/2, OM = r cos(θ/2) and AM = r sin(θ/2), so AB = 2r sin(θ/2). The area of triangle OAB is then 1/2 × AB × OM. Substituting these values and subtracting from the sector area gives the area of the segment, often left in terms of π and the square root of 3.

• OM = r cos(θ/2), where M is the midpoint of the chord.
• AM = r sin(θ/2), so chord AB = 2r sin(θ/2).
• Area of triangle OAB = 1/2 × AB × OM.

Many real objects are made by joining or cutting circles, sectors, triangles, squares and rectangles. To find the area of such a combination, first identify the simple shapes involved and decide whether their areas are to be added or subtracted. For example, the area swept by a rotating wiper or a clock hand is a sector; the region a tied animal can graze is a sector of a circle whose radius equals the rope length; the area between two consecutive ribs of an umbrella is one of several equal sectors; and a design on a round table cover may be a segment repeated several times. Always check the units and use the value of π given in the question.

• Break the figure into known shapes such as sectors, triangles and circles.
• Add areas of parts that are joined; subtract areas of parts that are removed.
• For n equal sectors, area of one sector = total area ÷ n.

The formulas of this chapter are all linked to the same circle of radius r. The circumference 2πr corresponds to 360°, the area πr² corresponds to 360°, and the sector of angle θ is simply the fraction θ/360 of each. This is why the same fraction θ/360 appears in both the arc length and the sector area formulas. The area of a segment is then the sector area minus the triangle area. Keeping this structure in mind makes the formulas easier to remember and helps you decide which one to use in a given problem.

• Arc length = θ/360 × circumference.
• Sector area = θ/360 × area of circle.
• Segment area = sector area − triangle area.