# What is the circumference and area of a circle with a diameter of 13 centimeters?

Oct 17, 2015

Formula for circumference of a circle:

$C = 2. \pi . r$

where,

r is the radius of the circle.

$r = \frac{\mathrm{di} a m e t e r}{2}$
$r = \frac{13}{2}$
$r = 6.5$ centimeters.

$C = 2. 3142.6 .5$
$C = 40.846$ centimeters

Formula for Area of a Circle:

$A = \pi . {r}^{2}$
$A = 3.142 . {\left(6.5\right)}^{2}$
$A = 132.7495$ Sq. centimeters

Oct 17, 2015

The circumference is $2 \pi r = 2 \pi \cdot \frac{13}{2} = 13 \pi \approx 40.84 c m$
The area is $\pi {r}^{2} = \pi {\left(\frac{13}{2}\right)}^{2} = \frac{169 \pi}{4} \approx 132.73 c {m}^{2}$

#### Explanation:

$\pi$ is defined as the ratio between the circumference of a circle and its diameter. $\pi$ is an irrational number, so it cannot be represented as a fraction or a repeating decimal.

Popular rational approximations for $\pi$ are $\frac{22}{7} = 3. \dot{1} 4285 \dot{7}$ and the less well known but really good $\frac{355}{113} \approx 3.1415929$.

So the circumference of our circle will be $\pi \cdot 13 c m = 13 \pi c m \approx 13 \cdot \frac{355}{113} \approx 40.84 c m$

The formula more commonly used is $2 \pi r$, where $r$ is the radius of the circle.

The formula for the area of a circle is $\pi {r}^{2}$, where $r$ is the radius.

In our case we have:

$\text{area} = \pi {r}^{2} = \pi {\left(\frac{13}{2}\right)}^{2} = \pi \left({13}^{2} / {2}^{2}\right) = \pi \left(\frac{169}{4}\right) \approx \frac{355 \cdot 169}{113 \cdot 4} = \frac{59995}{452} \approx 132.73 c {m}^{2}$

Why is the area $\pi {r}^{2}$?

You can divide a circle of radius $r$ into a number of segments and rearrange them head to tail to form a sort of bumpy parallelogram with long side roughly $\pi r$ and height $r$, therefore area $\pi r \cdot r = \pi {r}^{2}$.

This work better the more segments you have, but here's an illustration for just $8$ segments...