Area of Circle/Proof 3

Theorem
The area $A$ of a circle is given by:
 * $A = \pi r^2$

where $r$ is the radius of the circle.

Proof

 * [[File:Area Equal.jpg]]

Refer to the figure.

Construct a circle with radius r and circumference c, where its area is denoted by $C$.

Construct a triangle with height r and base c, where its area is denoted by $T$.

Lemma 2: $T\ge C$

 * [[File:Area Smaller.jpg]]

This will be proven by contradiction.

Assume $TT \implies \frac{hq}{2}>\frac{rc}{2}$

On the other hand, $\displaystyle 0C$.

It should be possible to construct a regular polygon with area $P$, where $C\frac{rc}{2}$.

Hence a contradiction is obtained.

Hence $\neg T>C$.

Hence $T\le C$.

Final Proof

 * $T\ge C$ (from Lemma 2)
 * $T\le C$ (from Lemma 3)
 * $\therefore T \mathop = C$.
 * $\therefore C \mathop = T \mathop = \pi r^2$ (from Lemma 1)