Question

Formula to find area of sector is
(a)${\displaystyle \frac{1}{2}}rl$
(b)${\displaystyle \frac{\pi r\theta }{{360}^{\circ }}}$
(c)${\displaystyle \frac{\pi r^2\theta }{{180}^{\circ }}}$
(d)$\pi r^2$

Answer 1

Hint: Here we will make use of area of a circle and some basic properties of circle geometry. Use the fact that ratio of area of a sector to the area of circle will be same as the ratio of central angle of sector to ${360}^{\circ }$ that is $${\displaystyle \frac{{\mathrm{A}}_s}{{\mathrm{A}}_c}}={\displaystyle \frac{\mathrm{C}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{A}\mathrm{n}\mathrm{g}\mathrm{l}\mathrm{e}}{{360}^{\circ }}}$$Here, $${\mathrm{A}}_s$$ is the area of sector and $${\mathrm{A}}_c$$ is the area of circle.

Complete step by step solution:
A sector is a portion of a circle which is bounded by the radius and the adjoining arc. The semi-circle is also expressed as a sector of a circle, and it covers half a portion of the circle. Generally, a sector is categorized into two parts which are a major sector and a minor sector. Major sector has a larger central angle which is greater than ${180}^{\circ }$ . The minor sector has an angle less than ${180}^{\circ }$.

The following is the schematic diagram of the circle having radius r and angle $\theta$.

If central angle of sector is $\theta$ and radius of circle is $r$ then Area of sector will be${\displaystyle \frac{\theta }{{360}^{\circ }}}\times \pi r^2$ because area of circle is $\pi r^2$ , where r is the radius of the circle.

Note: Here, the central angle should be taken in degrees always not in radians. If the central angle of the sector is given in radians then do not use 360 degrees in the denominator. First convert 360 degrees in radians. One other way, you can do is convert the given angle of the sector in degrees.