Answer:
[tex]A=7\pi\ cm^2-\frac{343}{6}\ cm^2[/tex]
Step-by-step explanation:
Area of plane figures
Being r the radius of a circle, the area of a sector defined by an angle [tex]\theta[/tex] is
[tex]A_s=\frac{\theta}{2} r^2[/tex]
If a is the repeated side of an isosceles triangle and [tex]\beta[/tex] is the angle they define, then the area of the triangle is
[tex]A_t=\frac{a^2}{2}sin\beta[/tex]
The figure shows a circle with radius of r=7 cm. The white area is equal to the area of the circle minus the blue area
The area of the circle is
[tex]A_c=\pi r^2=49\pi cm^2[/tex]
The blue area is the sum of the sector defined by the angle (360-150)=[tex]210^o[/tex] and the triangle. An angle of [tex]210^o[/tex] is equivalent to
[tex]\frac{\pi}{180^o}210^o=\frac{7}{6}\pi[/tex]
The area of the sector is
[tex]A_s=\frac{7}{12}\pi (7^2)=\frac{343\pi}{12}\ cm^2[/tex]
The area of the triangle with center angle 150^o is
[tex]A_t=\frac{(7)(7)}{2}sin150^o[/tex]
[tex]A_t=\frac{49}{2}\frac{1}{2}[/tex]
[tex]A_t=\frac{49}{4}\ cm^2[/tex]
The blue area is
[tex]\frac{49}{4}\ cm^2+\frac{343}{12}\pi\ cm^2[/tex]
Finally, the white area is
[tex]A=49\pi cm^2-(\frac{49}{4}\ cm^2+\frac{343}{12}\pi\ cm^2)[/tex]
[tex]A=\frac{245}{12}\pi\ cm^2-\frac{49}{4}\ cm^2[/tex]
