 Side (a)
in
Area
in²
Perimeter
in
Long diagonal (d)
in
Short diagonal (s)
in
in
in

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 do not only explain why the 6-sided polygon is so popular, but also how to correctly draw hexagon sides. 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 teaching you how to use the calculator to simplify any calculation involving this 6-sided shape.

## How many sides does a hexagon have? Exploring the 6-sided shape

It should come as no surprise that the hexagon (also known as the "6-sided polygon") has precisely six sides. This 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 not only the most common way to represent a 6-sided polygon but also the one most often found in nature. There will be a whole section dedicated to the important properties of the hexagon shape, but first we need to know the technical answer to: "What is a hexagon?" This will help us understand the tricks we can use to calculate the area of a hexagon without using the hexagon area formula blindly. These tricks involve using other polygons such as squares, triangles and even parallelograms.

## Hexagon definition, what is a hexagon?

In very much the same way an octagon is defined as having 8 angles, a hexagonal shape is technically defined as having 6 angles which conversely means that (as you could seen in the picture above) that the hexagonal shape is always a 6-sided shape. The angles of an arbitrary hexagon can have any value, but they all must to sum up to 720º (you can easily convert to other units using our angle conversion calculator).

In a regular hexagon, however, all the hexagon sides and angles have to have the same value. For the sides, any value is accepted as long as they are all the same. This means that, for a regular hexagon, calculating the perimeter is so easy that you don't even need to use the perimeter of a polygon calculator if you know a bit of maths. Just calculate:

`perimeter = 6 * side`,

where `side` refers to the length of any one side.

As for the angles, a regular hexagon requires that all angles are equal and the sum up to 720º which means that each individual angle must be 120º. This proves to be of the utmost importance when we talk about the popularity of the hexagon shape in nature. It will also be useful when we explain how to find the area of a regular hexagon since we will be using these angles to figure out which triangle calculator we should use, or even which rectangle we will use in another method.

## Hexagon area formula: how to find the area of a hexagon We will now have 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 of the sides and the center. It can be viewed as the height of the equilateral triangle formed 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 angles. For the regular hexagon these triangles are equilateral triangles. This 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 an octagon.

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 exactly 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, if you divide the hexagon in half (from vertex to vertex) you get 2 trapezoids, and you can calculate the area of the hexagon as the sum of both, using our trapezoid area calculator. 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! Check out the area of the right triangle calculator for help with the computations.

## Diagonals of a hexagon The total number of hexagon's 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 - The do not cross the central point. They are constructed joining two vertices leaving exactly one in between them. Their length is equal to `d = √3 * a`.

You should be able to check the statement about the long diagonal by visual inspection. However checking the short diagonals take a bit more ingenuity. But don't worry it's not rocket science, and we are sure that you can calculate it yourself with a bit of time; remember that you can always use the help of calculators such as the right triangle side lengths calculator.

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 a half of its height, which is also the apothem: `r = √3/2 * a`.

## How to draw a hexagon shape

Now we are going to explore a more practical and less mathematical world: how to draw a hexagon. For a random (irregular) hexagon the answer is simple: draw any 6-sided shape so that it is a closed polygon and you're done. But for a regular hexagon things are not so easy since we have to make sure all the sides are of the same length. To get the perfect result you will need a drawing compass. Draw a circle, and, with the same radius, start making marks along it. Starting at a random point and then making the next mark using the previous one as the anchor point draw a circle with the compass. You will end up with 6 marks, and if you join them with the straight lines, you will have yourself a regular hexagon. You can see a similar process on the animation above.

## The easiest way to find a hexagon side, area...

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, as well as including 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 centre to a vertex (Same as the radius of the hexagon),
• `Incircle radius`: Same as the apothem.

If you like the simplicity of this calculator we invite you to try our other polygon calculators such as the regular pentagon calculator or even 3-D calculators such as the pyramid calculator, triangular prism calculator, or the rectangular prism calculator.

## Hexagon tiles and real-world uses of the 6-sided polygon

Everyone loves a good real-world application, and hexagons are definitely one of the most used polygons in the world. Starting with human usages, the easiest (and probably least interesting) use is hexagon tiles for flooring purposes. The hexagon is an excellent shape because it perfectly fits with one another to cover any desired area. If you're interested in such a use, we recommend the flooring calculator and the square footage calculator as they are very good tools for this purpose. The next case is common to all polygons, but it is still interesting to see. In photography, the opening of the sensor almost always has a polygonal shape. This part of the camera called the aperture, and dictates many properties and features of the pictures taken by the camera. The most unexpected one is the shape of very bright (point-like) objects due to the effect called diffraction grating, and it is illustrated in the picture above. One of the most important uses of hexagons in the modern era, closely related to the one we've talked about in photography, is in astronomy. One of the biggest problems we experience when trying to observe distant stars is how faint they are in the night sky. That is because despite being very bright objects, they are so very far away that only a tiny fraction of their light reaches us; you can learn more about that in our luminosity calculator. On top of that, due to relativistic effects (similar to time dilation and length contraction), their light arrives on the Earth with less energy than it was emitted. This effect is called the red shift. The result is that we get a tiny amount of energy with a bigger wavelength than we would like. The best way to counteract this, is to build telescopes as big as possible. The problem is that making a one-piece lens or mirror bigger than a couple 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 mirrors 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.

## Honeycomb pattern - why the 6-sided shape is so prevalent in nature

The honeycomb pattern is composed of regular hexagons arranged side by side. They completely fill the entire surface they span, so there aren't any holes in between them. This honeycomb pattern appears not only in honeycombs (surprise!) but also in many other places in nature. In fact, it is so popular that one could say it is the default shape when conflicting forces are at play, and spheres are not possible due to the nature of the problem.

From bee 'hives' to rock cracks through organic chemistry (even in the build blocks of life: proteins), regular hexagons is 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 the full description of the importance and advantages of regular hexagons, we recommend watching the video above. To those who are hard-core readers, read on (you can check how fast you read with the reading speed calculator).

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 in 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 amongst these surface-filling polygons, which obviously makes it very efficient. A very interesting example in the video above 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 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 happens to also 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 visualising the benefits of a hexagon over other shapes, but is not the only way. In nature, as we have mentioned, there are plenty of example 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 Causeway)
• Eyes of insects
• ...
Bogna Haponiuk and Álvaro Díez

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