
The Rectangle’s Diagonal
A rectangle inscribed in a quarter circle with a radius of 1: what is the exact length of its diagonal AC?

Planet
On Polygon there are no improvised curves. Everything is made of polygons and solids: the mountains are pyramids, the trees are cones, the houses are rectangular boxes, and even the clouds are cubes floating placidly across the sky, sometimes in a checkerboard formation. When it rains, it rains tetrahedra, small and harmless.
The inhabitants, the Polygonians, measure angles and diagonals the way we check the temperature, and they greet each other by tracing shapes in the air. The only circle on the planet is the equator, and the Polygonians are rather proud of it: they treat it as a national monument, and heaven help anyone who calls it “just a curved line.”
Dord finds Polygon reassuring, although he complains that some of the triangles aren’t perfectly equilateral. Liz tried to improvise a little dance, but was politely informed that her steps had to trace a regular hexagon. M00N, being round, became an instant celebrity: everyone wants a selfie with her, the most famous moon on the planet of polygons.

A rectangle inscribed in a quarter circle with a radius of 1: what is the exact length of its diagonal AC?

With six matches, and without bending or breaking them, can you make four equilateral triangles? A classic from Polygon.

Nine dots arranged in a square and four straight lines drawn without lifting the pen: can you connect them all?

A rope as long as Earth’s equator, lengthened by just one yard: can an ant, a cat, or an elephant pass underneath it?