Area of Circle/Proof 3

Proof

 * Area-of-Circle-Proof-3.png

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

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

Final Proof
From Lemma $2$:
 * $T \ge C$

From Lemma $3$:
 * $T \le C$

Therefore:
 * $T \mathop = C$

and so from Lemma $1$:
 * $C \mathop = T \mathop = \pi r^2$