How to find the radius of a sector, when given only the angle and perimeter of the sector?

Question

I have been puzzling over this for a while and can't figure out how to find "r" without needing at least another variable.

Help would be much appreciated!

Answer 1

The perimeter is 40 cm, let's call it $P$. Let's call the arc length of the portion of the circle $s$. Then we have $$P = r + r + s = 2r + s = 40$$ One thing we will use is the formula for the circumference of a circle. In particular, the circumference of a circle is $$C=2\cdot\pi\cdot r$$ But here, we don't have a factor of just $2\pi$, since it's not a full circle, but $(72/360)$ of a full circle. Therefore, we have $$s=\frac{72}{360}\cdot 2\pi\cdot r $$ Substituting $s$ back into our perimeter equation above yields $$P=2r+\frac{72}{360}\cdot 2\pi\cdot r =40$$ Now we have a $1$-variable equation that can be solved. Can you take it from here?

Answer 2

Hint: If the angle were the full circle ($360^{\circ})$, the length of the arc would be $2\pi r$.

Instead, the angle is only $\tfrac{72}{360}=\tfrac15$ of the full circle, so the corresponding arc length is $\tfrac15(2\pi r) = \tfrac25 \pi r$.

So now you can easily write an expression for the full perimeter and finish the problem.