Shape

A shape is the form of an object or region defined by its boundaries and features. In mathematics, shapes are studied as sets of points in a plane or space that satisfy certain properties.

Two-dimensional shapes include polygons (triangles, quadrilaterals, pentagons, and other polygons), circles, and ellipses. They have boundaries and interiors. Key features are edges, vertices, and angles. Measurements include perimeter, the length of the boundary, and area, the size of the interior. Regular shapes have equal sides and angles. Convex shapes contain all line segments between any two points in the shape; concave shapes do not. Shapes can be classified by symmetry and angle properties, and by concepts of similarity and congruence.

Three-dimensional shapes, or solids, include polyhedra such as the cube and tetrahedron, as well as cylinders, cones, spheres, pyramids, and toroids. They have faces, edges, and vertices. They enclose volume and have surface area. Formulas compute these measures in terms of dimensions. Some properties, like the Euler characteristic V − E + F = 2 for convex polyhedra, relate numbers of vertices, edges, and faces.

Transformations such as translation, rotation, reflection, and scaling change the position or size of shapes. A transformation may preserve shape (congruence) or preserve shape with size (similarity). Symmetry describes invariance under certain transformations, such as reflection or rotation.

In topology, many properties are preserved under continuous deformations, focusing on features such as connectedness and the number of holes (genus). Shapes are a foundational concept in geometry, architecture, design, and the sciences.