We given formula for the length of arc : $\dfrac{2\pi r \theta }{360}$ where $\theta$ is angle subtended by an arc. Similarly, for area we have given fomula $\dfrac{\pi r^2 \theta}{360}$ where $\theta$ is angle subtended by an arc….
But I can't understand what are the rigorous proofs of these formulas and what are the rigorous definitionss of arc length and area of a sector. Please…also give me a name of a best book which can solve my questions.(This is both reference request and a question.)
thanks…
Best Answer
The formulas are valid when $\theta^\circ$ is the angle measured in degrees.
There is no rigorous proofs needed; what we need are the definitions of arc length, circumference, and area of a circle and sector: both the arc length corresponding to a subtended angle (whose vertex is the center of the circle), and the area corresponding to the resultant sector are simply fractions of the entire circle's circumference and area, respectively, equal to $\frac{\theta}{360}^{\text{th}}$ of the entire circle's circumference and area, respectively.
So, given a circle $O$ with radius $r$, we know its circumference is given by $2\pi r$. Given an angle $\theta$ measured in degrees, extended to the perimeter of the circle, then the length of the arc subtended by the angle is $\frac{\theta}{360}^{\text{th}}$ of the entire circle's circumference: $$L = \frac {\theta^\circ}{360^\circ}\cdot 2 \pi r = \frac {2 \pi r \theta^\circ}{360^\circ}$$
Likewise, given a circle $O$ with radius $r$, the area $A$ of a sector determined by $\theta$ (measured in degrees) is $\frac{\theta}{360}^{\text{th}}$ of the area of the entire circle:
$$A = \frac{\theta^\circ}{360^\circ}\cdot \pi r^2 = \frac{\pi r^2 \theta^\circ}{360^\circ}$$
See arc length and area of a sector of a circle for explicit definitions and additional explanations and images to aid in understanding.
Less detailed are the entries in Proof Wiki for area of a sector and arc length of sector. You'll see, they do things entirely in radians: $\theta $ is measured in radians, and so the fraction of circumference/area of a circle is given by $\dfrac{\theta}{2\pi}$ instead of $\dfrac{\theta^\circ}{360}.$