The given information is:
- The area of the sector of the circle is 31.2 cm^2.
- The exterior of the sector angle is 256°.
The area of the sector is given by the formula:
[tex]A=(\frac{\theta}{360\degree})*\pi r^2[/tex]
Where theta is the sector angle and r is the radius of the circle.
The whole circle measures 360°, so the sector angle can be found as follows:
[tex]\begin{gathered} \theta+256\degree=360\degree \\ \theta=360\degree-256\degree \\ \theta=104\degree \end{gathered}[/tex]
Now, replace theta and the area value in the formula to solve for r:
[tex]\begin{gathered} 31.2=\frac{104}{360}*\pi r^2 \\ \\ 31.2*\frac{360}{104}=\pi r^2 \\ \\ 108=\pi r^2 \\ \\ r^2=\frac{108}{\pi} \\ \\ r^2=34.4 \\ \\ r=\sqrt{34.4} \\ \\ r=5.9 \end{gathered}[/tex]
The radius of the circle is 5.9 cm.
