Area of Circle/Kepler's Proof

Proof
Let the circle of radius $r$ be divided into many sectors:


 * AreaOfCircleProof5.png

If they are made small enough, they can be approximated to triangles whose heights are all $r$.

Let the bases of these triangles be denoted:
 * $b_1, b_2, b_3, \ldots$

From Area of Triangle in Terms of Side and Altitude, their areas are:
 * $\dfrac {r b_1} 2, \dfrac {r b_2} 2, \dfrac {r b_3} 2, \ldots$

The area $\mathcal A$ of the circle is given by the sum of the areas of each of these triangles:

But $b_1 + b_2 + b_3 + \cdots$ is the length of the circumference of the circle.

From Perimeter of Circle:
 * $b_1 + b_2 + b_3 + \cdots = 2 \pi r$

Hence:

It needs to be noted that this proof is intuitive and non-rigorous.

Historical Note
This was the method used by when he was working on his Second Law of Planetary Motion.