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Class 10 · Chapter 11 · Mathematics

Areas Related to Circles Cheat Sheet

Formulae

Circumference of a circle

C=2πrC = 2\pi r

r = radius. Also C = πd, where d is the diameter.

Area of a circle

A=πr2A = \pi r^{2}

r = radius of the circle.

Area of a ring (annulus)

A=π(R2−r2)A = \pi\left(R^{2} - r^{2}\right)

R = outer radius, r = inner radius of the circular ring.

Length of an arc of a sector

l=θ360∘×2πrl = \frac{\theta}{360^\circ} \times 2\pi r

θ = central angle in degrees, r = radius.

Area of a sector

A=θ360∘×πr2A = \frac{\theta}{360^\circ} \times \pi r^{2}

θ = central angle of the sector in degrees.

Area of the major sector

A=360∘−θ360∘×πr2A = \frac{360^\circ - \theta}{360^\circ} \times \pi r^{2}

The part of the circle left when the minor sector of angle θ is removed.

Area of a segment

A=θ360∘×πr2−12r2sin⁡θA = \frac{\theta}{360^\circ} \times \pi r^{2} - \frac{1}{2} r^{2} \sin\theta

Area of the minor segment = area of sector − area of the triangle formed by the radii.

Definitions

Circumference
The distance around a circle, equal to 2πr.
Sector
The region enclosed by two radii and the arc between them.
Minor sector
The smaller sector, whose central angle is less than 180°.
Major sector
The larger sector, whose central angle is greater than 180°.
Segment
The region enclosed by a chord and the corresponding arc.
Minor segment
The segment cut off by a chord that subtends an acute central angle.
Arc
A part of the circumference of a circle.
Chord
A line segment joining two points on a circle.

Key Points

  • π is a constant; for calculations use π = 22/7 when the radius is a multiple of 7, otherwise π = 3.14.
  • The area of a sector is a fraction θ/360 of the area of the whole circle.
  • The length of an arc is the same fraction θ/360 of the circumference.
  • The angle in a complete circle is 360°, so a semicircle has angle 180° and a quadrant has angle 90°.
  • Area of a semicircle = ½πr² and area of a quadrant = ¼πr².
  • Area of a segment = area of the corresponding sector − area of the triangle formed by the two radii and the chord.
  • The distance covered by a wheel in one revolution equals its circumference.
  • The perimeter of a sector = 2r + length of the arc.