Defining Geometric Figures

What is a two-dimensional figure?

A two-dimensional figure, also called a plane or planar figure, is a set of line segments or sides and curve segments or arcs, all lying in a single plane. The sides and arcs are called the edges of the figure. The edges are one-dimensional, but they lie in the plane, which is two-dimensional.

The endpoints of the edges are called the vertices or corners. These points are zero-dimensional, but they also lie in the plane, which is two-dimensional. The most common figures have only a few edges, the curves are very simple, and there are no "loose ends" - that is, every vertex is the endpoint of at least two edges.

If all the edges are segments, every vertex is the endpoint of two sides, and no two sides cross each other, the figure is called a polygon.1 Polygons are classified according to the number of sides they have, which equals the number of vertices. Here are some names of polygons.

Polygons often divide the plane into two pieces, an inside and an outside. The inside part is called the region enclosed by the figure. The name of the figure is also commonly used for this region, and the area of the region is commonly called the area of the figure.

When two sides meet at a vertex, they form an angle. Actually they form two angles, one inside the figure, and one outside. The one inside is called the interior angle at that vertex, or simply the angle at that vertex.

Mathematically speaking, a triangle consists of three vertices and three sides only. The interior is not included. When you want to refer specifically to the interior of a figure that does not have a name of its own, you can call it "the region of the plane enclosed by the figure" or "the figurate region": for example, the "triangular region." When you calculate the "area of the triangle" you are really finding the area of the region enclosed by the triangle.

1 polygon - from Greek polus, "many," and gonia, "angle." Although a polygon is defined as a figure with many sides, the word really means that it has many angles.


What is a three-dimensional figure?

A three-dimensional figure, sometimes called a solid figure, is a set of plane regions and surface regions, all lying in three-dimensional space. These surface regions are called the faces of the figure. Each of them is two-dimensional. The arcs of curves that are the edges of the faces of the figure are called the edges of the figure. They are one-dimensional. The endpoints of the edges are called its vertices. They are zero-dimensional.

The most common three-dimensional figures have only a few faces, the surfaces are very simple, and there are no "loose ends" - that is, every vertex is the end of at least two edges, and at least two faces meet at every edge.

If all the faces are plane regions, every edge is the edge of two faces, every vertex is the vertex of at least three faces, and no two faces cross each other, the figure is called a polyhedron.2 Polyhedra are classified according to the number of faces. Here are some names of polyhedra.

Polyhedra often divide space into two pieces, an inside and an outside. The inside part is called the region enclosed by the figure. The name of the figure is also commonly used for this region, and the volume of the region is commonly called the volume of the figure.

When two planar faces come together at an edge, they form an angle. Actually they form two angles, one inside the figure, and one outside. The one inside is called the dihedral angle (dihedral means "having two faces") at that edge, or simply the angle at that edge.

Mathematically speaking, a tetrahedron consists of four triangular faces, six edges, and four vertices only. The interior is not included. When you want to refer specifically to the interior of a figure that does not have a name of its own, you can call it "the region of space enclosed by the figure" or the figurate solid: for example, the "tetrahedral solid." When you calculate the "volume of the figure" you are really finding the volume of the region enclosed by the figure, or the "figurate solid."

2 polyhedron - from Greek polus, "many," and hedra, "base" or "seat." A polyhedron is thus a figure with many bases, or faces.


A note on dimensions

A point, which is 0-dimensional, can lie on a line, in a plane, or in space. A line, which is 1-dimensional, can lie in a plane or in space. A plane, which is 2-dimensional, can lie in space, and space is 3-dimensional.

Similarly, curves are 1-dimensional, but can lie in higher-dimensional objects, and likewise for surfaces, which are 2-dimensional.

Solids, which are 3-dimensional, can only lie in space.