Areas Related To Circles Class 10 Notes Maths: Chapter 12

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Distance moved by a wheel in 1 rotation = Circumference of the wheel.
Number of rotation by a wheel in 1 minute = Distance moved by wheel in 1 minute / Circumference of wheel.

For a circle having radius r.
(i) Diameter = 2r
(ii) Circumference = 2πr
(iii) Area = πr2
(iv) Area of semi-circle =r²/2
(v) Area of a quadrant =r²/4
(vi) Perimeter of semi-circle = (πr + 2r)

If R and r are the radii of two concentric circles such that R > r then,The area enclosed between the two circles = πR² – πr² = π(R² – r²)
A segment of a circle is the region bounded by a chord and the arc subtended by the chord

$$If \space a \space sector\space of \space a \space circle \space of \space radius\space r\space contains\space an\space angle\space of\space \theta°, then$$

$$\\(i)\space Length\space of\space the\space arc\space of\space the\space sector =\frac{\theta}{{360°}}×2\pi r=\frac{\theta}{{360°}}×(Circumference\space of\space the\space circle)$$

$$\\(ii) Perimeter\space of\space the\space sector\space = 2r+\frac{\theta}{{360°}}×2\pi r$$

$$\\ (iii) Area\space of\space the\space sector\space = \frac{\theta}{{360°}}×\pi r^2=\frac{\theta}{{360°}}×(Area of the circle)$$

$$\\ (iv) Area\space of\space the\space minor\space segment = Area\space of\space the\space corresponding\space sector – Area\space of\space the\space corresponding\space triangle$$

$$\\ =\frac{\theta}{{360°}}×\pi r^2– r^2 \space sin\space\frac{\theta}{{2}}cos\space\frac{\theta}{{2}}$$

$$\\ =\set{\frac{\theta}{{360°}}×\pi – \space sin\space\frac{\theta}{{2}}cos\space\frac{\theta}{{2}}}r^2$$

$$\\ =\set{\frac{\theta}{{360°}}×\pi – \space \space\frac{1}{{2}}sin\space{\theta}{}}r^2$$

$$\\(iv) Area\space of\space the\space major\space segment\space = Area\space of\space the\space circle\space – Area\space of\space the\space minor\space segment$$

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