If you divide the circumference by $2$, does it equal the area divided by the radius? That is, do you have

$C/2 = A/r$

for any circle?

  • 3
    $\begingroup$ Yes, both are $\pi r$. $\endgroup$ – Thomas Andrews Jul 30 '15 at 23:37

Yes. $$ \frac{C}{2} = \frac{2 \pi r}{2} = \pi r.$$

Similarly, $$\frac{A}{r} = \frac{\pi r^2}{ r} = \pi r.$$


A good way to view this intuitively is to imagine a circle to be composed of small rings.

Now unroll the rings all the rings into a flat surface, you get a triangle of base $C = 2\pi r$ and height $r$ so area $A=\frac{Cr}{2}$


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