Set in the world of Flatland, A. Square is a plane form living in a two dimensional reality. His place in the social hierarchy is in the middle class, and he describes effectively the other classes within his world. The dangerous Women, or single lines, the dull yet powerful Isosceles Triangles, or soldiers and laymen, the Equilateral Triangle, or tradesmen, the Nobility class, or Polygon class, and the Circles, or Priests, all combine to make a social order within Flatland. Their existence is complete with class strife, discrimination, alteration, regulation, wars, and power struggles, just as any other society in existence.

When A. Square dreams of another world, Lineland, he is struck by a need to explain to the inhabitants the error of their one dimensional viewpoints. Although A. Square tries repeatedly to explain the second dimension to the inhabitants, the King of Lineland is not convinced. A. Square, believing he is superior to the Lineland inhabitants, eventually loses his temper, and is threatened by the Lineland subjects.

Waking from his dream, A. Square is soon visited by a stranger claiming to be of a third dimension. Just as he felt the need to teach Lineland individuals the error of their views, the stranger, Sphere, attempts the same with A. Square. His efforts through analogy, description, explanation, and action all fail, however, as A. Square believes his world to be superior to all others. However, when A. Square threatens Sphere, he is thrust into the third dimension to see for himself the reality of a solid existence.

A. Square learns that each millennium, a member of the land of Spaceland comes to Flatland to explain the third dimension. He and the Sphere listen, however, as the Circles of Flatland plan to imprison or kill anyone during this millennium change who claims a visit from such a creature. When Sphere enters the hall of Circles, he is threatened by death, but disappears again into the third dimension. Upon returning to Spaceland, A. Square tries in vain to force Sphere to show him other dimensions, but Sphere, not knowing of these dimensions, becomes angry and A. Square is sent back to Flatland to teach others of the third dimension.

On his return, A. Square has a vision of Pointland, the land of no dimensions. In the vision, the single existence in Pointland is contemplating his own perfection, and Sphere explains that to the Point, there are no other existences. As the only creature, it believes itself to be the center of all things, and the definition of perfection. When A. Square attempts to convince it otherwise, the creature hears only its own voice, since it remains unaware of any other being.

On his return to Flatland, A. Square attempts to convince his countrymen of the existence of the third dimension. As foretold, however, he is imprisoned for his views, and remains behind bars for the rest of his days. A. Square hopes his writings will inspire other individuals in a later time to pursue other dimensions, as he now realizes his knowledge is lost on his hierarchical society.