Proof 5: Ellipsis area

Imagine a circle with a radius of exactly 1, $r=1$ (called a unit circle). Circle area is $A_\text{circle}=\pi r^2$ (see Proof 4), so such a unit circle’s area will be:

$$A_\text{unit circle}=\pi\,1^2=\pi $$

  • Stretch it by 3 sideways and by 3 upwards/downwards and the area triples twice: $\pi\cdot3 \cdot 3=\pi \,3^2$.
  • Stretch it by 4 both sideways and upwards/downwards, and the area is made four times larger twice: $\pi \cdot 4 \cdot 4=\pi \,4^2$.
  • In general, stretch it by a value $r$, both sideways and upwards/downwards and you are back at the usual circle area formula: $\pi \cdot r \cdot r=\pi\,r^2$.

If you don’t stretch equally much sideways and upwards/downwards, but instead, say, 3 times sideways and 4 times upwards/downwards, then you have a new shape – not a circle – with a new area 3 times and 4 times larger: $\pi \cdot 3 \cdot 4$.

This new shape is a ‘flattened’ circle, called an ellipsis. In general, the pattern shows that the values you stretch with, let’s call them $r_1$ and $r_2$, are just multiplied onto the unit circle area:

$$A_\text{ellipsis}=\pi r_1r_2\quad_\blacksquare$$