I have a perimeter exercise of a circular sector, but my result is different. Where have I been wrong?
The exercise is:
With the formula: $\frac{\angle{O}}{360}2\pi r + 2r$
I have the circular sector of the biggest circle is equal to $4\pi + 16$ and the smallest is equal to $3\pi + 12$, then i substract and i get $\pi + 4$.
But according to the guide, must be equal to $7\pi + 4$, what is wrong?
$\endgroup$ 31 Answer
$\begingroup$The circumference $C$ of a circle of radius $r$ is given by $C=2\pi r$
The sector of said circle with a central angle $\alpha$ has an arc length of $\frac{\alpha C}{360}$
Thus the arc length of the circles is given by $\frac{2\pi r\alpha}{360}$
$\alpha=90$ is fixed here as the angle is right, then we have $r=8$ for the outer circle and $r=6$ for the inner circle.
So the arc length of the outer circle is $\frac{1440\pi}{360}=4\pi$ and of the inner circle is $\frac{1080\pi}{360}=3\pi$.
The perimeter is these two arc lengths plus twice the difference of the two radii (which is $8-6=2$ - see your diagram for where these come from), and so the total perimeter is $4\pi+3\pi+4=7\pi+4$.
Let me know if any of this doesn't make sense
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