Question

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

Answer 1

Formula for circumference of a circle:

`C = 2.pi.r`

where,

r is the radius of the circle.

`r = (diameter) / 2`
`r = 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 . (6.5)^2`
`A = 132.7495` Sq. centimeters

Answer 2

The circumference is `2pi r = 2pi*13/2 = 13pi ~~ 40.84cm`
The area is `pi r^2 = pi (13/2)^2 = (169pi)/4 ~~ 132.73cm^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 `22/7 = 3.dot(1)4285dot(7)` and the less well known but really good `355/113 ~~ 3.1415929`.

So the circumference of our circle will be `pi * 13cm = 13pi cm ~~ 13*355/113 ~~ 40.84cm`

The formula more commonly used is `2pir`, where `r` is the radius of the circle.

The formula for the area of a circle is `pir^2`, where `r` is the radius.

In our case we have:

`"area" = pi r^2 = pi (13/2)^2 = pi (13^2/2^2) = pi (169/4) ~~ (355*169)/(113*4) = 59995 / 452 ~~ 132.73cm^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 * r = pi r^2`.

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