CBSE Class 10 Mathematics Areas Related to Circles Notes Set 10

Circles: Circumference and Area

For a circle of radius r the circumference is \( 2\pi r \) and the area is \( \pi r^2 \); the diameter is 2r. A wheel of circumference C covers a distance C in one revolution, so the number of revolutions in a journey is the distance divided by C. Two circles touch internally when the distance between their centres is the difference of the radii, and externally when it is the sum.
The area between two concentric circles of radii R and r (a ring) is \( \pi(R^2-r^2) \). On a clock, the minute hand turns \( 6^\circ \) per minute and the hour hand \( 30^\circ \) per hour.

Sector, Semicircle and Segment

A sector with central angle \( \theta \) is the fraction \( \frac{\theta}{360} \) of the circle. So arc length \( =\frac{\theta}{360}\times2\pi r \), area \( =\frac{\theta}{360}\times\pi r^2 \), and the perimeter of the sector is the arc plus two radii. A semicircle is the case \( \theta=180^\circ \): its area is \( \frac12\pi r^2 \) and its perimeter \( \pi r+2r \).
A segment is the part of a circle cut off by a chord. The minor segment has area equal to the sector area minus the area of the triangle formed by the chord and the two radii; the major segment is the whole circle minus the minor segment.

Working with Combined Figures

To find the area of a shaded region, express it as sums and differences of simple pieces such as squares, triangles, semicircles and sectors. Check units before adding, and use the value of \( \pi \) that the question specifies. Equal-area problems (such as a circle equal in area to two given circles) are solved by adding the areas first and then taking the radius of the new circle.