Square Inscribed in Circle is greater than Half Area of Circle

Theorem

A square inscribed in a circle has an area greater than half that of the circle.

Proof

Let $ABCD$ be a square inscribed in a circle.

Let $EFGH$ be a square circumscribed around the same circle.

We have that:

$ABCD$ is twice the area of the triangle $ADB$.
$EFGH$ is twice the area of the rectangle $EFBD$.

From Area of Rectangle, the area of $EFBD$ is $ED \cdot DB$.

From Area of Triangle in Terms of Side and Altitude, the area of $ADB$ is $\dfrac 1 2 \cdot ED \cdot DB$.

Thus:

$EFGH = 2 \cdot ABCD$

But the area of $EFGH$ is greater than the area of the circle around which it is circumscribed.

Therefore half of the area of $EFGH$ is greater than half of the area of the circle around which it is circumscribed.

Therefore the area of $ABCD$ is greater than half of the area of the circle within which it is inscribed.

$\blacksquare$

Historical Note

This result is used in Euclid's The Elements, specifically Book $\text{XII}$, in several places. However, it is never extracted and proved separately as a lemma; the result is merely stated without comment after the first time it is demonstrated.

On $\mathsf{Pr} \infty \mathsf{fWiki}$ it has been determined as being an important enough result to be proved in its own right.