Solid Terminology

From flat shapes to solids

A square is a two-dimensional shape: it has length and width. A cube is a solid, with three dimensions: length, width and height. Solids occupy space. A box, a tin and a ball can help us picture different solids, although real objects may have extra details that the mathematical shapes do not.

Some solids have only flat surfaces. These are called polyhedra: their faces are polygons, such as squares, rectangles or triangles. Other solids have curved surfaces.

Faces, surfaces, edges and vertices

The surface is the outside boundary of a solid. A face is an individual flat part of that surface. A cube has six faces, each a square; a sphere has a curved surface but no flat faces.

For a polyhedron, an edge is a straight line segment where two faces meet. A vertex is a corner point where edges meet. The plural of vertex is vertices.

Imagine running a finger over a cube. Across the middle of a square you are touching a face. Along the join between two squares you are following an edge. At a corner you reach a vertex. Altogether, a cube has 6 faces, 12 edges and 8 vertices, including the ones hidden from view.

Curved solids need a little more care. The circular rim of a cylinder is a curved boundary, not a straight edge joining two vertices. Keeping flat faces and curved surfaces separate makes these shapes easier to describe precisely.

Cubes, cuboids and prisms

A cube has six equal square faces. All its edges have the same length.

A cuboid has six rectangular faces, with opposite faces matching. Its length, width and height need not be equal. A cereal box is a useful model. A cube is a special cuboid because a square is a special rectangle.

A prism has two matching, parallel polygonal ends. Between them, its cross-section stays the same. A cross-section is the flat shape revealed by a slice through a solid. To recognise a prism, imagine taking slices parallel to its ends: each slice has the same shape and size.

Prisms are named after this cross-section. A triangular prism has triangular ends and, in the usual right prism, three rectangular joining faces. A hexagonal prism has hexagonal ends. Cubes and cuboids are also prisms, with square and rectangular cross-sections respectively.

Cylinders: a constant circular cross-section

A cylinder has two equal, parallel circular faces and one curved surface joining them. A straight-sided food tin is a useful model.

Like a prism, it has a constant cross-section when sliced parallel to its ends. The difference is that this cross-section is a circle, not a polygon, and the joining surface is curved. A cylinder is therefore not a polyhedron.

Pyramids and cones: narrowing to a point

A pyramid has a polygonal base. Its other faces are triangles that meet at a single point called the apex. The base gives the pyramid its name.

A square-based pyramid has one square face and four triangular faces. A triangular-based pyramid, also called a tetrahedron, has four triangular faces altogether: the base and three joining faces.

Unlike a prism, a pyramid narrows towards its apex. Slices parallel to its base become smaller as they approach the top.

A cone also narrows to a point, but it has a circular base and a curved surface rather than triangular faces. A cone is not a pyramid because its base is not a polygon and its joining surface is curved.

Spheres: curved all the way round

A sphere is a round, ball-shaped solid. It has one continuous curved surface, no flat faces, no edges and no vertices.

To distinguish the curved solids, look for their defining features: a cylinder has two circular faces, a cone has one circular base and a point, and a sphere has neither a flat base nor a point.

Essential terminology

• Face: an individual flat surface.
• Surface: the outside boundary; it may be flat or curved.
• Edge of a polyhedron: a straight join between two faces.
• Vertex: a corner; plural vertices.
• Polyhedron: a solid made entirely of polygonal faces.

Recognise the solids

Key distinctions

• Prism: stays the same size along its length. Pyramid: narrows towards an apex.
• Name prisms by their cross-section and pyramids by their base.
• Cubes and cuboids are prisms. Cylinders, cones and spheres are not polyhedra.
• A triangular-based pyramid is a tetrahedron.