Areas Related to Circles Class 10: notes and important questions

Basic results (circle of radius \(r\))

• Circumference \(=2\pi r\); area \(=\pi r^2\).
• For a sector with central angle \(\theta\) (in degrees):
  • Length of arc \(l=\dfrac{\theta}{360^\circ}\times2\pi r\)
  • Area of sector \(=\dfrac{\theta}{360^\circ}\times\pi r^2=\dfrac12\,l\,r\)
  • Perimeter of sector \(=2r+l\)
• Major sector area \(=\pi r^2-\) minor sector area (its angle is \(360^\circ-\theta\)).
• Segment (the region between a chord and its arc): minor segment area \(=\) area of sector \(-\) area of \(\triangle OAB\).
  • \(\theta=60^\circ\): triangle is equilateral, area \(=\dfrac{\sqrt3}{4}r^2\)
  • \(\theta=90^\circ\): area \(=\dfrac12r^2\)
  • \(\theta=120^\circ\): area \(=\dfrac{\sqrt3}{4}r^2\) (height \(\tfrac r2\), base \(r\sqrt3\))
• Major segment area \(=\pi r^2-\) minor segment area.

Worked example 1

Find the area of a sector of angle \(40^\circ\) in a circle of radius 21 cm \(\left(\pi=\frac{22}{7}\right)\).

\(\dfrac{40}{360}\times\dfrac{22}{7}\times441=\dfrac19\times1386=154\ \text{cm}^2\).

Worked example 2

A chord of a circle of radius 10 cm subtends \(60^\circ\) at the centre. Find the minor segment area (\(\pi=3.14,\ \sqrt3=1.73\)).

Sector \(=\dfrac16\times3.14\times100=52.33\); triangle \(=\dfrac{1.73}{4}\times100=43.25\); segment \(\approx9.08\ \text{cm}^2\).

Worked example 3

The tip of a 12 cm minute hand moves for 25 minutes. How far does it travel?   Angle \(=25\times6^\circ=150^\circ\); arc \(=\dfrac{150}{360}\times2\times3.14\times12=31.4\) cm.

Handy facts

• The minute hand turns \(6^\circ\) per minute; the hour hand turns \(0.5^\circ\) per minute.
• Sector areas of the same circle are in the ratio of their angles. For a fixed angle, sector area is proportional to \(r^2\) and arc length to \(r\).

Common errors

• Adding the triangle to the sector instead of subtracting it to get the minor segment.
• Using the diameter in place of the radius.
• Forgetting the two radii when finding the perimeter of a sector.
• Using \(\pi=3.14\) when the question says \(\frac{22}{7}\), or rounding too early.

Board-exam tips

• Syllabus limit: segment problems use central angles of 60°, 90° and 120° only.
• Write the formula first, then substitute. This earns the method mark even if the arithmetic slips.
• If the answer is in terms of \(\pi\) and \(\sqrt3\), leave it exact unless a value is given.