# 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:

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

## Proof 11: Sphere and ellipsoid volume

A sphere can be cut into slices. Each slice is almost a disk, or a very flat cylinder, except for the curved edge. If we…

## Proof 13: Ellipsis equation

The circle equation (see Proof 12) can be rearranged: $$(x-c_1)^2+(y-c_2)^2=r^2\\\frac{(x-c_1)^2}{r^2}+\frac{(y-c_2)^2}{r^2}=1\\\left(\frac{x-c_1}{r}\right)^2+\left(\frac{y-c_2}{r} \right)^2 =1$$ The $x-c_1$ and $y-c_2$ distances are being ‘squeezed’ by $r$, so to say.…

## Proof 6: Prism and cylinder volume

In a room with 3 metres to the ceiling, there are 3 cubic metres above each square metre floor regardless of the shape. The 3…

## Proof 4: Circle and circle sector area

A pizza slice’s edge is the pizza’s radius $r$. The slice is almost a triangle if it wasn’t for the rounding. That rounding has a…

## Proof 17: Power rules of algebra

Definitions: The pattern looks as if reducing the exponent means removing an $a$ term: \vdots\\a^4=a\cdot a \cdot a \cdot a\\a^3=a\cdot a \cdot a\\ a^2=a\cdot a…