Area of Circle/Proof 3/Lemma 1

Lemma for Area of Circle: Proof 3

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 $T$.

Then:

$T = \pi r^2$

Proof

\(\ds T\)

\(=\)

\(\ds \frac {r c} 2\)

Area of Triangle in Terms of Side and Altitude

\(\ds \)

\(=\)

\(\ds \frac {r 2 \pi r} 2\)

Perimeter of Circle

\(\ds \)

\(=\)

\(\ds \pi r^2\)

$\blacksquare$