How to Find the Diagonal of a Hexagon

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The diagonals of hexagons may be calculated by understanding hexagon structure and correlating the side of a hexagon to its radius. Keep reading to find out how to do the math.

Properties of Regular Hexagons

A hexagon is a six-sided polygon or 6-gon. The word hexagon comes from the Greek hex, meaning six, and gonia, meaning corner or angle.

Properties of regular hexagons:

  • interior angles of 120 degrees
  • the total of the interior angles of a hexagon is 720 degrees
  • each side and the interior angles are equal to each other
  • there are no curved sides
  • all lines connect to make the shape closed

An irregular hexagon has six unequal sides. A convex hexagon has no angles pointing inward. A concave hexagon has an angle greater than 180 degrees (pointing inward).

Diagonals of Hexagons

To find the diagonals of hexagons, use the formula:

n (n-3)/2, where n is the number of sides of a polygon.

For a hexagon, n = 6, and 6 (6-3) / 2 equals nine diagonals.

A regular hexagon shape has a radius that equals the side length. This creates six triangles. Recall that a radius of a hexagon is the center point of the hexagon to one of its corners.

Also recall that a diagonal is a line joining two opposite corners of a straight-sided shape. For regular hexagons, the nine diagonals form into six equilateral triangles.

Determining the Length of Diagonals in Hexagons

Since the nine diagonals form into six equilateral triangles, and the radius equals the side length, this makes it straightforward to determine the length of each diagonal line. If one side of the hexagon is known, then all sides are known, and the diagonals are calculated using the following basic steps:

Step 1: Determine the Length of One Side of the Hexagon

All the sides are equal in a regular hexagon. If the length of one side is known, then all are. The known, or given, is labeled as "g" (given side).

Step 2: Calculate Diagonal of Hexagon

Note the equation for finding the diagonal of a regular hexagon:

d (diagonal) = 2g (given side)

Multiply the known or given side of the hexagon by two. The product is the length of the diagonal of a regular hexagon.

For example, if the given side equals 10 meters, then the diagonal is: 2(10 meters), or 20 meters.

Diagonals of an Irregular Hexagon

There is no standard formula for finding the diagonals of irregular hexagons.

Although you can calculate the number of diagonals in an irregular hexagon, finding the diagonal measurement of an irregular would require splitting the hexagon into triangles. However, if they are not right triangles, there is not a format for finding the length of the interior side, the diagonal. The Pythagorean Theorem applies only to right triangles.

If each side and angle were given along with the area, then the diagonals could be determined; however, it is unlikely so many variables would be specified in a problem.

The Hexagon in Nature

A beehive is one of the most easily identifiable hexagon structures in nature. In a beehive, there are interconnected hexagons, and this structure was found to be good for packing as it leaves no empty space within the hive. For the same reason, soap bubbles, when lining up, create hexagonal shapes.

When water spins at high velocity, it takes the shape of a hexagon. Likewise, there is a perpetual storm-like cloud in the shape of a hexagon at Saturn’s north pole.

A carbon ring is a hexagon shape with a carbon at each corner. Dragonfly eyes have a hexagonal shape and so do the patterns on a tortoise shell.


About the Author

Rosann Kozlowski is currently a freelance writer and tutor. She has a Master's Degree in Chemistry from the University of Oregon and has previously worked in the pharmaceutical industry and has taught at the middle school, high school, and college levels.

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