Hexagon Calculator | 6 - Sided Polygon (2025)

Welcome to the hexagon calculator, a handy tool when dealing with any regular hexagon. The hexagon shape is one of the most popular shapes in nature, from honeycomb patterns to hexagon tiles for mirrors – its uses are almost endless. Here we explain not only why the 6-sided polygon is so popular but also how to draw hexagon sides correctly. We also answer the question "what is a hexagon?" using the hexagon definition.

With our hexagon calculator, you can explore many geometrical properties and calculations, including how to find the area of a hexagon, as well as teach you how to use the calculator to simplify any analysis involving this 6-sided shape.

It should be no surprise that the hexagon (also known as the "6-sided polygon") has precisely six sides. This fact is true for all hexagons since it is their defining feature. The length of the sides can vary even within the same hexagon, except when it comes to the regular hexagon, in which all sides must have equal length.

We will dive a bit deeper into such shape later on when we deal with how to find the area of a hexagon. For now, it suffices to say that the regular hexagon is the most common among 6-sided polygons and the one most often found in nature.

As we've already mentioned, the regular hexagon must have all sides of equal length and all its internal angles must be equal. Each side length is equally valid (as long as it's shared by all 6 sides!), so computing the perimeter of a hexagon is so simple you don't even need the perimeter of a polygon calculator. Just calculate:

• perimeter = 6 × side

The angles of an arbitrary hexagon can have any value, but they all must sum up to 720º (you can easily convert them to other units using our angle conversion calculator). As the regular hexagon requires that all angles are equal, it follows that each individual angle must be 120º. This fact proves to be of the utmost importance when we talk about the popularity of the hexagon shape in nature. It will also be helpful when we explain how to find the area of a regular hexagon. We'll use it also to find the area formula for regular hexagons.

We will now take a look at how to find the area of a hexagon using different tricks. The easiest way is to use our hexagon calculator, which includes a built-in area conversion tool. For those who want to know how to do this by hand, we will explain how to find the area of a regular hexagon with and without the hexagon area formula. The formula for the area of a polygon is always the same no matter how many sides it has as long as it is a regular polygon:

• area = apothem × perimeter / 2

Just as a reminder, the apothem is the distance between the midpoint of any side and the center. You can view it as the height of the equilateral triangle formed by taking one side and two radii of the hexagon (each of the colored areas in the image above). Alternatively, one can also think about the apothem as the distance between the center, and any side of the hexagon since the Euclidean distance is defined using a perpendicular line.

If you don't remember the formula, you can always think about the 6-sided polygon as a collection of 6 triangles. For the regular hexagon, these triangles are equilateral triangles. This fact makes it much easier to calculate their area than if they were isosceles triangles or even 45 45 90 triangles as in the case of a square.

For the regular triangle, all sides are of the same length, which is the length of the side of the hexagon they form. We will call this a. And the height of a triangle will be h = √3/2 × a, which is the exact value of the apothem in this case. We remind you that √ means square root. Using this, we can start with the maths:

• A₀ = a × h / 2
• = a × √3/2 × a / 2
• = √3/4 × a²

Where A₀ means the area of each of the equilateral triangles in which we have divided the hexagon. After multiplying this area by six (because we have 6 triangles), we get the hexagon area formula:

• A = 6 × A₀ = 6 × √3/4 × a²

• A = 3 × √3/2 × a²

• = (√3/2 × a) × (6 × a) /2

• = apothem × perimeter /2

We hope you can see how we arrive at the same hexagon area formula we mentioned before.

If you want to get exotic, you can play around with other different shapes. For example, suppose you divide the hexagon in half (from vertex to vertex). In that case, you get two trapezoids, and you can calculate the area of the hexagon as the sum of them. You could also combine two adjacent triangles to construct a total of 3 different rhombuses and calculate the area of each separately. You can even decompose the hexagon in one big rectangle (using the short diagonals) and 2 isosceles triangles!

Feel free to play around with different shapes and calculators to see what other tricks you can come up with. Try to use only right triangles or maybe even special right triangles to calculate the area of a hexagon!

The total number of hexagon diagonals is equal to 9 – three of these are long diagonals that cross the central point, and the other six are the so-called "height" of the hexagon.
Our hexagon calculator can also spare you some tedious calculations on the lengths of the hexagon's diagonals. Here is how you calculate the two types of diagonals:

• Long diagonals – They always cross the central point of the hexagon. As you can notice from the picture above, the length of such a diagonal is equal to two edge lengths:

  D = 2 × a

• Short diagonals – They do not cross the central point. They are constructed by joining two vertices, leaving exactly one in between them. Their length is equal to d = √3 × a.

Another pair of values that are important in a hexagon are the circumradius and the inradius. The circumradius is the radius of the circumference that contains all the vertices of the regular hexagon. The inradius is the radius of the biggest circle contained entirely within the hexagon.

• Circumradius: to find the radius of a circle circumscribed on the regular hexagon, you need to determine the distance between the central point of the hexagon (that is also the center of the circle) and any of the vertices. It is simply equal to R = a.

• Inradius: the radius of a circle inscribed in the regular hexagon is equal to half of its height, which is also the apothem: r = √3/2 × a.

The hexagon calculator allows you to calculate several interesting parameters of the 6-sided shape that we usually call a hexagon. Using this calculator is as simple as it can possibly get with only one of the parameters needed to calculate all others and includes a built-in length conversion tool for each of them.

We have discussed all the parameters of the calculator, but for the sake of clarity and completeness, we will now go over them briefly:

• Area – 2–D surface enclosed by the hexagon shape;
• Side Length – Distance from one vertex to the next consecutive one;
• Perimeter – Sum of the lengths of all hexagon sides;
• Long Diagonal – Distance from one vertex to the opposite one;
• Short Diagonal – Distance between two vertices that have another vertex between them;
• Circumcircle radius – Distance from the center to a vertex (Same as the radius of the hexagon); and
• Incircle radius – Same as the apothem.

The result is that we get a tiny amount of energy with a longer wavelength than we would like. The best way to counteract this is to build telescopes as enormous as possible. The problem is that making a one-piece lens or mirror larger than a couple of meters is almost impossible, not to talk about the issues with logistics. The solution is to build a modular mirror using hexagonal tiles like the ones you can see in the pictures above.

Making such a big mirror improves the angular resolution of the telescope, as well as the magnification factor due to the geometrical properties of a "Cassegrain telescope". So we can say that thanks to regular hexagons, we can see better, further, and more clearly than we could have ever done with only one-piece lenses or mirrors.

Did you know that hexagon quilts are also a thing?? Discover more with Omni's hexagon quilt calculator!

From bee 'hives' to rock cracks through organic chemistry (even in the build blocks of life: proteins), regular hexagons are the most common polygonal shape that exists in nature. And there is a reason for that: the hexagon angles. The 120º angle is the most mechanically stable of all, and coincidentally it is also the angle at which the sides meet at the vertices when we line up hexagons side by side. For a full description of the importance and advantages of regular hexagons, we recommend watching this video.

The way that 120º angles distribute forces (and, in turn, stress) amongst 2 of the hexagon sides makes it a very stable and mechanically efficient geometry. This is a significant advantage that hexagons have. Another important property of regular hexagons is that they can fill a surface with no gaps between them (along with regular triangles and squares). On top of that, the regular 6-sided shape has the smallest perimeter for the biggest area among these surface-filling polygons, which makes it very efficient.

A fascinating example in this video is that of the soap bubbles. When you create a bubble using water, soap, and some of your own breath, it always has a spherical shape. This result is because the volume of a sphere is the largest of any other object for a given surface area.

However, when we lay the bubbles together on a flat surface, the sphere loses its efficiency advantage since the section of a sphere cannot completely cover a 2D space. The next best shape in terms of volume-to-surface area ratio also happens to be the best at balancing the inter-bubble tension that is created on the surface of the bubbles. We are, of course, talking of our almighty hexagon.

Bubbles present an interesting way of visualizing the benefits of a hexagon over other shapes, but it's not the only way. In nature, as we have mentioned, there are plenty of examples of hexagonal formations, mostly due to stress and tensions in the material. We cannot go over all of them in detail, unfortunately. We can, however, name a few places where one can find regular hexagonal patterns in nature:

• Honeycombs;
• Organic compounds ;
• Stacks of bubbles;
• Rock formations (like Giant's);
• Eyes of insects;
• ...

In a hexagon, the apothem is the distance between the midpoint of any side and the center of the hexagon. When you imagine a hexagon as six equilateral triangles that all share a vertex at the hexagon's center, the apothem is the height of each of these triangles.

The answer is √3, that is, approximately 1.73. This is because of the relationship apothem = ½ × √3 × side. When we plug in side = 2, we obtain apothem = √3, as claimed.

The answer is 3√3/2, that is, approximately, 2.598. To arrive at this result, you can use the formula that links the area and side of a regular hexagon. It reads area = 3√3/2 × side², so we immediately obtain the answer by plugging in side = 1.