- How many sides does a hexagon have? Exploring the 6-sided shape
- Hexagon definition, what is a hexagon?
- Hexagon area formula: how to find the area of a hexagon
- Diagonals of a hexagon
- Circumradius and inradius
- How to draw a hexagon shape
- The easiest way to find a hexagon side, area...
- Hexagon tiles and real-world uses of the 6-sided polygon
- Honeycomb pattern - why the 6-sided shape is so prevalent in nature
Welcome to the hexagon calculator which is 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 not only explain why the 6-sided polygon is so popular, but also how to draw the 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 you can learn 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 "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 from one to another except for the regular hexagon in which all sides must have equal length. We will dive a bit deeper into such shape later on, in particular regarding 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. This tricks will involve using other polygons such as the squares, triangles and even parallelograms.
Hexagon definition, what is a hexagon?
In very much the same fashion as an octagon is defined as having 8 angles, the 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 together have to sum up to 720º which 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,
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 a 120º one. 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 explaining how to find the area of a regular hexagon since we will be using this angles to figure out which triangle calculator should we use, or even to know the properties of the rectangle that we will use in another method.
Hexagon area formula: how to find the area of a hexagon
We will see now how to find the area of a hexagon using different tricks. The easiest way is to use our hexagon calculator, that includes a built-in area conversion tool. For those who want to know how they could 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 radius 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 centre 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 the 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 height of a triangle will be
h = √3/2 * a which is exactly the value of the
apothem. 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
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
And you can see that in this manner we arrive at the same hexagon area formula we mentioned before.
If you want to get exotic, you can play 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 to adjacent triangles to construct a total of 3 different rhombus of which you can calculate the area separately. You can even decomponse the Hexagon in one big rectangle (using the short diagonals) and 2 isosceles triangles!
Feel free to play around with different shapes and calculators and 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 also called the "height" of the hexagon. Our hexagon calculator can also spare you some tedious calculations on the lengths of the hexagon's diagonals. There are two types of these 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.
One can easily check, visually, the size of the long diagonal by simply noting that it is composed by two segments of the same size as 2 of the sides. However checking the short diagonals take a bit more ingenuity. But don't worry it's not rocket physics 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.
Circumradius and inradius
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 inside 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 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.
In the perfect world, we would have a drawing compass and we would get a perfect result. Just draw a circumference 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 for 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 a 6-sided shape that we usually call hexagon. The usage is as simple as it can possibly get with only one of the parameters needed to calculate the rest and 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 opposed 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 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) 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 very good tool for this purpose.
The next use 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 aperture 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 the one in astronomy. In astronomy, one of the biggest problems 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 far away that only a minimal amount 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 the logistics. The solution is to build a modular mirror using hexagon 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 the regular hexagons we can see better, further and more clear 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're filling the entire surface they span, that is, there aren't any wholes in between them. This honeycomb pattern appears not only in honeycombs (surprise!) but 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 'houses' to rock cracks through organic chemistry (even including the build blocks of life: proteins), the regular hexagon 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 a full description of the importance and the advantages of regular hexagons, we recommend watching the video. To those who are hard-core readers, we will try to explain briefly, and we recommend them checking how fast they 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 advantage of the regular hexagon is that it belongs to the group of polygons that can fill a surface with no gaps in between them (regular triangle, square and hexagon). 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 that you can see in the video and experiment on your own 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 a section of a sphere cannot cover completely a 2D space. The next best shape in terms of volume to surface area happens to be also the best at balancing the inter-bubble tensions that are created on the surfaces of the bubbles. Of course, we are talking of our almighty hexagon.
The bubbles present an interesting way to visualize the benefits of a hexagon over other shapes, but is not the only one. 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. However we can name a few places where one can find regular hexagonal patterns in:
- Organic compounds
- Stacks of bubbles
- Rock formations (like Giant's Causeway)
- Eyes of insects