Class 10 Maths Notes Chapter 11 — Areas Related to Circles#

Key terms#

  • Sector — the region between an arc and the two radii to its ends. The smaller one is the minor sector, the larger one the major sector.
  • Segment — the region between a chord and its arc. The smaller one is the minor segment, the larger one the major segment.
  • Angle of the sector — the angle $\theta$ that the two radii make at the centre.
  • For a minor sector of angle $\theta$, the major sector has angle $360° - \theta$.

OAB with the arc AB is the minor sector. The region between chord AB and arc AB is the minor segment.

Formulas#

For a circle of radius $r$ and a sector of angle $\theta$ (in degrees):

$$\text{Length of arc} = \frac{\theta}{360°} \times 2\pi r$$

$$\text{Area of sector} = \frac{\theta}{360°} \times \pi r^2$$

$$\text{Area of segment} = \text{Area of sector} - \text{Area of triangle}$$

Also useful:
Area of major sector = $\pi r^2$ − Area of minor sector
Area of major segment = $\pi r^2$ − Area of minor segment
Perimeter of a sector = $2r$ + length of arc

Area of the triangle in a segment#

The triangle is $\triangle OAB$, with $OA = OB = r$.

Angle $\theta$ Triangle Area
$60°$ Equilateral $\dfrac{\sqrt{3}}{4} r^2$
$90°$ Right-angled at $O$ $\dfrac{1}{2} r^2$
$120°$ Draw $OM \perp AB$: $OM = \dfrac{r}{2}$, $AB = \sqrt{3}\,r$ $\dfrac{\sqrt{3}}{4} r^2$

Real-life sectors#

  • Minute hand: turns 360° in 60 minutes, so 6° per minute. In 5 minutes it sweeps a sector of 30°.
  • Horse tied at a corner of a square field: it grazes a quadrant ($\theta = 90°$) of radius equal to the rope.
  • $n$ equal parts (umbrella ribs, brooch diameters): each sector has angle $\frac{360°}{n}$.
  • Wipers, lighthouse beams: each is a sector with the given angle and radius.

Common mistakes#

  • Using the diameter as $r$.
  • Mixing units, for example a radius in cm and an answer in m².
  • Using $\pi = \frac{22}{7}$ when the question says $\pi = 3.14$, or the other way round.
  • Forgetting to subtract the triangle when the question asks for a segment, not a sector.
  • Taking the minor angle for a major sector. Use $360° - \theta$.

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