In geometry, the vertex (vertices) refers to a point at which two or more edges, rays, or lines intersect. Vertices are critical in different shapes and figures, including polyhedra, polygons, and angles. A triangle, for instance, has three vertices, where its edges intersect, while a cube has eight vertices where the edges intersect. Understanding vertices is particularly crucial for American students who are preparing for standardized tests such as the SAT, ACT, and GRE, where a good understanding of geometric principles can improve performance. From identifying vertices in polygons to solving intricate geometry problems, an understanding of this concept builds a strong mathematical foundation for achieving success in competitive exams.

What are Vertices?

A vertex is considered a special point of a mathematical object and is defined as a location where two or more straight lines or edges meet. These points, also sometimes referred to as corners, are generally denoted by uppercase letters such as A, B, C, etc.

In general, we can refer to vertices as the

• The intersection point of two or more rays or the arms of an angle,
• The intersection point of the adjacent edges of a polygon,
• The intersection point of the edges of a solid figure

The most common examples of vertices are the tip of the cone, the corners of a cube or cuboid, the corners of a square, the rectangles and triangles in 2-dimensional figures, etc.

Vertices for 2d and 3d Shapes

Different geometric figures will have a different number of corners, or vertices. For example, a square has four vertices, a triangle has three, and a cone has one. So let's do a brief study on this by categorising figures into 2D and 3D.

• 2-dimensional figures

As we already know, a 2-dimensional shape has only 2 dimensions: length and width. Some 2-dimensional figures include squares, rectangles, triangles, and circles. In 2D shapes, the vertex is defined as the point of intersection of two sides of the shape. These two sides intersect to form an angle, and thus a vertex is formed. You can have a look at the number of vertices in different 2D shapes below.

• 3-dimensional figures 

As we know, a 3-dimensional shape has 3 dimensions: length, width, and height. Some 3-dimensional figures include cubes, cuboids, pyramids, and spheres. In 3D shapes, a "vertex" is defined as the point of intersection of two edges of the shape. In a 3-dimensional shape, multiple sides and edges can intersect from different dimensions at one point. Each intersection point forms a vertex. You can have a look at the number of vertices in different 3D shapes below,

What is the Difference Between Edges and Vertices?

The key differences between edges and vertices are as follows:

Do Circles have Vertices?

No, Circles do not have Vertices. Why? This is because as per the definition, vertex is the point of intersection of straight lines and in case of circles we do not have any straight lines that intersect to form vertex. 

This doesn't mean circles do not have Edges in any sense. Circles do have an edge which forms the boundary of the shape. The length of the edge gives the circumference of the circle.

Summary

Let's the summarize entire discussion:

• Vertex or Vertices (plural for “vertex”) are the points where two straight lines interest or come together to form a point.
• An edge is defined as the link between the vertices or interface between two faces.

Solved Example

Example 1. List down the vertices of the given figure.

Solution 1. In the given figure, we have, \(A, B, C, D, E, F, G, and H\) as vertices.

Example 2. Mark the vertices of the given figure.

Solution 2. The vertices of the given figure can be marked as,

Example 3. Name the figures, \(2D\) and \(3D\) which have no vertices.

Solution 3. Figures with no vertices are listed as follows:

\(In 2D: Circle, oval\).

\(In 3D: Sphere, Cylinder.\)

The concept of vertices is a basic one in geometry that is very important in 2D and 3D figures. Whether it is naming the corners of polygons or points where edges meet in solid figures, vertices are used to construct the foundation of geometric concepts. For American students gearing up for standardized tests such as the SAT, ACT, and GRE, a firm understanding of vertices can immensely upgrade problem-solving abilities and improve test scores. From identifying vertices of simple shapes to breaking down intricate figures, this understanding provides a robust foundation for cracking competitive exams. Set to upgrade your skills? Experience interactive learning and practice with the Testbook App to lead the pack in your prep journey!