Origami makes it possible to obtain constructions impossible with straightedge and compass alone, such as doubling the cube or trisecting the angle (see La Géométrie de la règle et du compas, Bibliothèque Tangente 78, 2022). The most emblematic of the construction problems of Antiquity, squaring the circle, that is, constructing a square of area π from a line segment of length 1, remains out of reach via origami.
But it is fun to try to obtain a "good approximation" of it.
Haga, Haga, Haga, and there's pi!
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Start from a square ABCD.
By bringing point D onto the midpoint of [AB], Haga's first theorem (see the article "Haga's fold") gives us the point E on [BC] such that CE = 1/3.
