Diagonals in Geometry: Shapes & Formula

Instructor: Karin Gonzalez

Karin has taught middle and high school Health and has a master's degree in social work.

In this lesson, you will learn the definition of a diagonal in geometry and see some examples in shapes. You will also learn the formula to determine how many diagonals are in a shape. Following the lesson is a brief quiz.

What is a Diagonal in Geometry?

Before we get to the definition of a diagonal of geometry, we need to understand the components that make up the definition:

  • A diagonal line, in informal terms, is simply a line that is at a slope.
  • A line segment is a section of a line that has two defined endpoints.
  • A polygon is a flat plane figure with at least three sides and angles. Usually, when we refer to a polygon, it has five or more sides and angles.
  • A polyhedron is the three-dimensional counterpart of a polygon. It also usually has five or more sides. But the minimum is again three.
  • Lastly, vertices are the points of a shape.

Now that we have all of that covered, a diagonal in geometry is a diagonal line segment that connects two nonconsecutive vertices of either a polygon or a polyhedron.

One more thing: based on the definition that a diagonal needs to connect two nonconsecutive vertices, it would be impossible to have a diagonal in a triangle or triangular pyramid.

Images of Diagonals in Geometry

If you roll through one of these without stopping completely, you may get a ticket. That's right - a stop sign is an example of a polygon. In the image, you can see examples of diagonals in geometry in the stop sign.

Note that each of the three diagonals are connected by nonconsecutive vertices of the stop sign.
Some diagonals in a hexagon

A square is a polygon. A cube is what you call a three-dimensional square, which is a polyhedron. Informally, we would call this a box.

Here is a cube with two diagonals indicated in blue and red, connected by nonconsecutive vertices
Some diagonals in a cube

Formula for Diagonals

If n is the number of sides of the polygon or polyhedron, then the formula to find the number of diagonals is:

n (n-3)/2

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