shapes

Shapes are geometric figures defined by their boundaries in two- or three-dimensional space. In geometry, shapes are studied for their properties, relations, and measurements. Two-dimensional shapes lie in a plane and are described by their edges and angles, while three-dimensional shapes enclose volume.

Two-dimensional shapes include polygons such as triangles, quadrilaterals, pentagons, and other multi-sided figures, as well as non-polygonal curves such as circles and ellipses. Polygons are characterized by a finite sequence of straight sides, vertices, and interior angles. They can be regular, with equal sides and angles, or irregular; they can be convex, with all interior angles less than 180 degrees, or concave, having indentations. The perimeter is the boundary length, and the area is the measure of enclosed space.

Three-dimensional shapes include polyhedra with flat faces, edges, and vertices (examples: cube, tetrahedron, octahedron, dodecahedron, icosahedron) and solids with curved surfaces such as spheres, cylinders, cones, and torus. Polyhedra are often categorized by the number of faces and by convexity. The surface area measures the boundary area, and the volume measures enclosed space.

Shapes are studied under geometry transformations: translations, rotations, reflections, and combinations known as rigid motions. Congruence preserves shape and size; similarity allows scaling. Symmetry refers to invariance under some transformation. Shapes can tessellate a plane when copies fill the plane without gaps.

Shapes appear in mathematics, design, architecture, engineering, art, and nature. The concept of shape interacts with topology, which studies properties preserved under continuous deformations.