Question

calcule a área de um segmento circular de um círculo cujas medidas do raio e do ângulo central são 5 cm e 150°,respectivamente.

Answer 1

$\mathrm{F\acute{o}rmula\ da\ \acute{a}rea\ do\ segmento\ circular:}\\\\A=\dfrac{r^2}{2}(\alpha-\mathrm{sen}\,\alpha),\text{ para }\alpha\text{ em radianos.}\\\\\mathrm{Primeiramente\ vamos\ converter\ }150^{\circ}\text{ em radianos.}\\\\\dfrac{x}{\pi}=\dfrac{150^\circ}{180^\circ}\longrightarrow{x}=\dfrac{150^\circ\pi}{180^{\circ}}\longrightarrow{x}=\dfrac{5\pi}{6}\\\\\\\text{Substituindo os valores na }\mathrm{f\acute{o}rmula\ da\ \acute{a}rea\ do\ segmento\ circular:}$

$A=\dfrac{r^2}{2}(\alpha-\text{sen}\,\alpha)\\\\\\A=\dfrac{(5\,cm)^2}{2}\left(\dfrac{5\pi}{6}-\text{sen}\left(\dfrac{5\pi}{6}\right)\right)\\\\\\A=\dfrac{25\,cm^2}{2}\left(\dfrac{5\pi}{6}-\dfrac{1}{2}\right)\\\\\\A=\dfrac{25\,cm^2}{2}\left(\dfrac{5\pi-3}{6}\right)\\\\\\\boxed{A=\dfrac{125\pi\,cm^2-75\,cm^2}{12}}\\\\\\\text{Considerando }\pi=3{,}14:\\\\A=\dfrac{125\pi\,cm^2-75\,cm^2}{12}\\\\\\A=\dfrac{125(3{,}14)\,cm^2-75\,cm^2}{12}\\\\\\A=\dfrac{392{,}5\,cm^2-75\,cm^2}{12}\\\\\\A=\dfrac{318{,}5\,cm^2}{12}$

$\boxed{A\approx26{,}54\,cm^2}$