$\pi$ is irrational

How to use it

1. 1Any length, area or volume that is a non-zero rational multiple of $\pi$ — a circumference $2\pi r$, an area $\pi r^2$, a volume $\frac{4}{3}\pi r^3$ with $r$ rational — is irrational.
2. 2To argue it, divide by the rational factor: if $4\pi r^2$ were rational then dividing by the rational $4r^2$ would make $\pi$ rational.
3. 3Use it to answer 'is this quantity rational?' questions about circles and spheres without computing anything.

Step-by-step

1. 1
   $A=\pi r^2=\pi {\left(\frac{3}{2}\right)}^2=\frac{9\pi }{4}$

   Why: The radius is rational, so the whole of the area apart from $\pi$ is rational.

2. 2
   If $\frac{9\pi }{4}$ were rational, then $\frac{4}{9}\times \frac{9\pi }{4}=\pi$ would be rational

   Why: Multiplying a rational by the non-zero rational $\frac{4}{9}$ leaves a rational.

3. 3
   $\pi$ is irrational, so $\frac{9\pi }{4}$ is irrational