Area of Circle/Proof 3

Theorem
The area $A$ of a circle is given by the formula $A=\pi r^2$, where $r$ is the radius of the circle.

Proof

 * [[File:Area Equal.jpg]]

Circle with radius r and circumference c, a triangle with height r and base c.


 * $\displaystyle T=\frac{rc}{2} = \frac{r*2r\pi}{2} = \pi r^2$

T is the area of the triangle

Triangle area is smaller than circle area

 * [[File:Area Smaller.jpg]]

Assume the triangle area is smaller than circles area.

It should be possible to construct a regular polygon that has an area greater than the triangles but smaller than the circles.

For any given polygons area holds


 * $\displaystyle P=\frac{hq}{2}$

where q is the circumference of the polygon and h is the height of any given triangular part of it and P is the area. From the assumption it should be $\displaystyle P>T \implies \frac{hq}{2}>\frac{rc}{2}$

but $\displaystyle h<r$ and $\displaystyle q<c$ which implies $\displaystyle \frac{hq}{2}<\frac{rc}{2}$

Contradiction, the triangle area cannot be smaller than the circles area

Triangle area is greater than circle area

 * [[File:Area Greater.jpg]]

Assume triangles area is greater than the circles area.

It should be possible to construct a regular polygon that has an area greater than the circles but smaller than the triangles.

For any given polygons area holds


 * $\displaystyle P=\frac{hq}{2}$

where q is the circumference of the polygon and h is the height of any given triangular part of it and P is the area.

From the assumption it should be $\displaystyle P\frac{rc}{2}$

Contradiction.

Both are false so the areas must be the same.