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

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 $T$.

Lemma $1$

$T = \pi r^2$

$\Box$

Lemma $2$

$T \ge C$

$\Box$

Lemma $3$

$T \le C$

$\Box$

Final Proof

From Lemma $2$:

$T \ge C$

From Lemma $3$:

$T \le C$

Therefore:

$T \mathop = C$

and so from Lemma $1$:

$C \mathop = T \mathop = \pi r^2$

$\blacksquare$

Historical Note

The was determined by Archimedes in his Measurement of a Circle.

Sources

• 1944: R.P. Gillespie: Integration (2nd ed.) ... (previous) ... (next): Chapter $\text I$: $\S 1$. Area of a Circle
• 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.5$: Archimedes (ca. $\text {287}$ – $\text {212}$ B.C.)
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): indirect proof
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): indirect proof