Please prove: arc-length proportional to radius

[zaydeshap]

Two circles, radii R1 and R2.

In each circle, draw two radii with central angle A.
Prove that the lengths of arc subtended by the same central angle A are proportional to the radii R1, R2.

I want to use the result in an elementary class, but not take it for granted, even though it appears obvious.

 

[Denis]

You could always cut out the 2 circles, then lay one on top of the other:
the "class" will probably "see" that no proof is really required...get my drift?

 

[fasteddie65]

You could use the formula s = rø for arc length.

S1 = R1 • A
S2 = R2 • A

S1/S2 = R1 • A/R2 • A = R1/R2

I'm not sure how "elementary" this class is, but this proof is quite basic.

 

[pka]

If the radius is \(\displaystyle r\) and the centeral angle is \(\displaystyle \theta\) and the arc length is \(\displaystyle s\)
the use the ratio \(\displaystyle \frac{2\pi r}{2\pi}=\frac{s}{\theta}\)