Given:
The length of the arc PQ=12 cm
The angle of the sector POQ is 60 degrees.
To find the area of the sector and perimeter of the sector:
Using the length of an arc formula,
[tex]\begin{gathered} l=\frac{\theta}{360}\times2\pi r \\ 12=\frac{60}{360}\times2\times\frac{22}{7}\times r \\ 12=\frac{22}{21}\times r \\ r=\frac{12\times21}{22} \\ r\approx11.455\operatorname{cm} \end{gathered}[/tex]
Using the formula of area of the sector,
[tex]\begin{gathered} A=\frac{\theta}{360}\times\pi r^2 \\ =\frac{60}{360}\times\frac{22}{7}\times(11.455)^2 \\ =68.76\operatorname{cm} \end{gathered}[/tex]
Therefore, the area of the sector is approximately 68.76 square cm.
Using the formula of the perimeter of the sector,
[tex]\begin{gathered} P=2r+l \\ =2(11.455)+12 \\ =22.91+12 \\ =34.91 \end{gathered}[/tex]
Therefore, the perimeter of the sector is approximately 34.91 cm.
