How did squinches and pendentives allow a dome to rest over a square room?

To support a dome over a square room, architects must bridge the transition from a cubic base to a circular foundation. Both pendentives and squinches serve this purpose by filling in the upper corners of the square space.12

Pendentives

A pendentive is a triangular segment of a spherical surface. Four of these inverted spherical triangles are placed at the corners of a square room. Their apexes rest on the room's piers, effectively conducting the weight and forces of the dome downward. The sides of the pendentives join to form arches over the four faces of the cube, while their upper bases meet to create a complete circular foundation upon which the dome rests.134

Squinches

A squinch is a device used to fill in the corners of a square or polygonal room to provide a support for a dome. This can be achieved through several methods:2
* Corbelling: Projecting successive courses of masonry beyond the one below.2
* Arches: Building one or more arches diagonally across the corner.2
* Niches: Constructing a niche with a half-dome in the corner.2
* Vaulting: Filling the corner with a small conical vault.2