‌A‌ ‌vertex is a mathematical word for a corner.‌ Most geometrical shapes, whether two or three dimensional, possess vertices. For instance, a square has four vertices, which are its four corners. A vertex can also refer to a point in an angle or in a graphical representation of an equation. The word vertex comes from Latin for the crown of a head.

In geometry, if two line segments intersect, ‌the point where the two lines meet is called the vertex.‌ This is true, regardless if the lines cross or meet at a corner. Because of this, ‌angles also have vertices.‌ An angle measures the relationship of two line segments, which are called rays and which meet at a specific point. Based on the above definition, you can see that this point is also the vertex of an angle.

‌A 2D shape, such as a triangle, is composed of two parts – edges and vertices.‌ The ‌edges‌ are the lines that make up the boundary of the shape. Each point where two straight edges intersect is a vertex. A triangle has three edges – its three sides. It also has three vertices, which are each corner where two edges meet. Polygons are 2D shapes with at least three edges and vertices. With polygons, the number of sides is equivalent to the number of vertices. Different shapes have different terminology to describe them. For example, when a polygon has 4 sides it is called a quadrilateral.

Polygons have a convenient relationship between the number of sides (or vertices) and their interior angles. The sum of the interior angles can be found with this formula:

You can also see from this definition that ‌some two-dimensional shapes do not have any vertices.‌ For example, circles and ovals are made from a single edge with no corners. Since there are no separate edges intersecting, these shapes have no vertices. A semi-circle also has no vertices, because the intersections on the semi-circle are between a curved line and a straight line, instead of two straight lines.

‌Vertices are also used to describe points in three-dimensional shapes.‌ Three-dimensional objects are composed of three different parts. Take a cube: each of its flat sides is called a ‌face.‌ Each line where two faces meet is called an edge. Each point where two or more edges meet is a vertex. A cube has six square faces, twelve straight edges, and eight vertices where three edges meet. In other words, ‌each of the cube's corners is a vertex.‌ As with two-dimensional objects, some three-dimensional objects – such as spheres – do not have any vertices because they do not have intersecting edges.

The relationship between the faces and the vertices of a 3-D shape can be represented using Euler’s formula for polyhedrons, a very influential mathematical theorem. Given the number of faces (F), number of vertices (V), and the number of edges (E), it can be represented as follows:

The proof of this relationship is somewhat complicated, but it is super cool! We can look at more complex shapes like an octahedron – with 8 faces, 6 vertices, and 12 edges – or a tetrahedron – with 4 triangular faces, 4 vertices, and 6 edges to demonstrate this formula, but it will work for triangular prisms, hexagonal prisms, cuboids, and any 3D objects with straight edges.

‌Vertices are also used in algebra.‌ A ‌parabola‌ is a graph of an equation that looks like a giant letter "U." The equations that produce parabolas are called ‌quadratic equations,‌ and are variations on the formula:

‌A parabola has a single vertex‌ -- either at the bottom point of the "U," if the parabola opens upwards -- or at the top point of the "U," if the parabola opens downwards, like an upside down "U." For instance, the bottom point of the graph of the equation ‌y‌ = ‌x‌² is located at the point (0,0). The graph rises on either side of this point. So (0,0) is the vertex of the graph of ‌y‌ = ‌x‌².

The formula:

gives the x-coordinate of the vertex.