Formula Sheet – Area related to Circles – Class X

  1. Perimeter (circumference) of a circle with diameter d (d = 2r, where r is the radius) is given by C = \pi d = 2\pi r
  2. Perimeter of semicircle with radius r = 2r + \pi r = r(\pi+ 2)
  3. Area of a circle with radius r is given by A =\pi r^2
  4. Area of semicircle of radius r=\dfrac{\pi r^2}{2}
  5. Area of a ring whose outer and inner radii are R and r respectively =\pi(R^2-r^2)=\pi(R+r)\ (R-r)
  6. If two circles touch internally then the distance between their centres is equal to the difference of their radii
  7. The distance moved by a rotating wheel in one revolution is equal to the circumference of the wheel.
  8. The number of revolutions completed by a rotating wheel in one minute =\dfrac{Distance\ moved\ in\ one\ minute}{Circumference\ of\ the\ wheel}
  9. Length of an arc which substends an angle of \theta^o at the centre =\dfrac{2\pi r\theta^o}{360^o}=\dfrac{\pi r\theta^o}{180^o}
  10. Sector of a circle is a region enclosed by an arc of a circle and its two bounding radii.

(i) Area of sector OACBO=\dfrac{\pi r^2\theta^o}{360^o}
(ii) Perimeter of sector OACBO=2r+\dfrac{2\pi r^2\theta^o}{360^o}

11. Minor sector : A sector of a circle is called a minor sector if the minor arc of the circle is a part of its boundary. In the above figure minor sector is OACB.

12. Major sector : A sector of a circle is called a major sector, if the major arc of the circle is a part of its boundary. In the above figure, OADB is the major sector.

13. The sum of the arcs of major and minor sectors of a circle is equal to the circumference of the circle.

14. The sum of the areas of major and minor sectors of a circle is equal to the area of the circle.

15. The area of a sector is given by A=\dfrac{1}{2}lr, where =\left(\dfrac{\theta r}{180^o}\times \pi\right)

16. Angle described by minute hand in 60 minutes =360^o .
\therefore angle described by minute hand in one minute
=\left(\dfrac{360}{60}\right)=6^o

Thus, the minute hand rotate through an angle of  6^o in one minute.

17. Angle described by hour hand in  12 hours =360^o .

\therefore angle described by hour hand in 1 hour

=\left(\dfrac{360}{12}\right)^o=30^o

Angle described by hour hand in one minute

=\left(\dfrac{30}{60}\right)^o=\dfrac{1}{2}^o

Thus, hour hand rotates through \dfrac{1}{2}^o in 1 minute.

18. A segment of a circle is the region bounded by an arc and a chord, including the arc and the chord.

19. Minor segment : If the boundary of a segment is a minor arc of a circle, then the corresponding segment is called a minor segment. In the figure, segment PQR (the are which is shaded) is a minor segment.

20. Major segment : A segment corresponding a major arc of a circle is known as the major segment. In the figure above segment PQSP is a major segment.

21. Area of minor segment PRQS

=\dfrac{\pi r^2\theta^2}{360}-\dfrac{1}{2}r^2\sin\theta

22. Area of major segment PSQ

=\pi r^2 – area of minor segment PRQ.

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