Welcome to the hexagon calculator. 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 will not only explain why the 6-sided polygon is so popular but also how to draw the hexagon sides while we answer the question 'what is a hexagon?' Using the hexagon definition.
A hexagon has six sides and six corresponding angles. Each angle is 120 degrees and the sum of the angles is 720 degrees. A wooden hexagon made from six different pieces of wood will follow this rule. Cutting a 60-degree angle on each end of all six pieces results in six pieces of wood that will fit together and form a hexagon. Apr 24, 2017 - A hexagon is a shape with six sides. Using the correct equation, you can find the degree of each of the interior angles, or the angles inside the.
This being a hexagon calculator we will explore as well the geometrical properties and calculations including how to find the area of a hexagon as well as explaining 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 (a.k.a. '6-sided polygon') has precisely 6 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 such as the, and even. Hexagon definition, what is a hexagon? In very much the same fashion as an is defined as having 8 angles, the hexagonal shape is technically defined as having 6 angles which conversely means that (as we saw before) 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.
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 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 a 120º one.
How To Get Correct Hexagon Shape Out Of Paper
This will prove 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 to use, or even to know the properties of the that we will use in another method. Hexagon area formula: how to find the area of a nhexagon We will see now how to find the area of a hexagon using different tricks. The easiest way is to use our calculator, that includes a built-in 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 is always the same no matter how many sides it has as long as it is a regular polygon: Area = Apothem. Perimeter /2 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. This makes it much easier to calculate their area than if they were or even as in the case of the octagon.
For the regular triangle, all sides are the same length, which is the length of the side of the hexagon they form. We will call this a. And will be h = √3/2. a which is exactly the value of the Apothem. Using this we can start with the maths: A₀ = a. h / 2 = a. √3/2.
a / 2 = √3/4. a² 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. Feel free to play around with different shape and calculators and see what other tricks you can come up with. Can you use only or maybe even to calculate the area of a hexagon? Check out the 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 = 2a. Short diagonals - also called the height of the triangle, these diagonals do not cross the central point. Their length is equal to d = √3. a.
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. 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 the same length. Ideally, 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 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 a straight line, 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 The hexagon calculator allows you to calculate several interesting parameters of 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 tool for each of them. 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 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 and the 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 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, 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. On top of that due to relativistic effects (similar to and ), their light arrives on the Earth with less energy than it was emitted. This effect is called. All in all the result is that we get a tiny amount of energy and with a bigger 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 meter 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. Making such big mirrors improves the of the telescope as well as the 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 clear than we could have ever done with only one-piece lenses or mirrors. Honeycomb pattern and why the 6-sided shape is so prevalent in nature The honeycomb pattern composed of regular hexagons arrange side by side filling the entire surface the spam without any whole 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, 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 will recommend them checking how fast the read with the The way that 120º angles distribute amongst 2 of the sides the (and in turn the ) exerted on one the other one makes it a very stable and mechanically 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. So those are the main reasons you can find hexagons in many places. Just to name a few, one can find regular hexagonal patterns in:. Honeycombs. Organic compounds.
Stacks of bubbles. Rock formations (like ). Eyes of insects.
Explanation: If the apothem is and the question requires us to solve for the length of one of the sides, the problem can be resolved through the use of right triangles and trig functions. As long as one angle and one side length is known for a right triangle, trig functions can be used to solve for a mystery side. In the previous image, the side of interest has been labeled as. Keep in mind that is actually half the length of one side of the regular hexagon. In order to solve for, the first step is to solve for the measure of an internal angle of the hexagon.
This angle has been marked in the image. This can be solved for by using: where is the number of sides of the hexagon. In this problem,.
This measure of is the measure of the entire angle. Keep in mind that the drawn triangle is actually bisecting the internal angle.
This means that the angle of interest in the triangle is actually. Now that we have the apothem and one of the angles, we can use trig functions to solve for. Using SOH CAH TOA, we realize that this problem would require us to use the tan function where the ratio would be, or for this problem's data,. Now must be multiplied by because it's the length of half of the total length of one side of the hexagon.
Therefore, the final answer is. Explanation: This is a regular hexagon; therefore, when we draw in the diagonals, we will create identical equilateral triangles as shown by the figure: Now, the given line segment also serves as the height to the newly created equilateral triangles. This height also splits an equilateral triangle into two congruent triangles.
Recall that triangles have side lengths that are in the ratio of. Substitute in the given height into the ratio in order to find the length of the base of the triangle. The length of the side of the hexagon is twice the length of the base. Substitute in the value of the length of the base to find the side length of the hexagon. Explanation: This is a regular hexagon; therefore, when we draw in the diagonals, we will create identical equilateral triangles as shown by the figure: Now, the given line segment also serves as the height to the newly created equilateral triangles.
This height also splits an equilateral triangle into two congruent triangles. Recall that triangles have side lengths that are in the ratio of. Substitute in the given height into the ratio in order to find the length of the base of the triangle. The length of the side of the hexagon is twice the length of the base. Substitute in the value of the length of the base to find the side length of the hexagon.
Explanation: This is a regular hexagon; therefore, when we draw in the diagonals, we will create identical equilateral triangles as shown by the figure: Now, the given line segment also serves as the height to the newly created equilateral triangles. This height also splits an equilateral triangle into two congruent triangles. Recall that triangles have side lengths that are in the ratio of.
Substitute in the given height into the ratio in order to find the length of the base of the triangle. The length of the side of the hexagon is twice the length of the base. Substitute in the value of the length of the base to find the side length of the hexagon.
Explanation: This is a regular hexagon; therefore, when we draw in the diagonals, we will create identical equilateral triangles as shown by the figure: Now, the given line segment also serves as the height to the newly created equilateral triangles. This height also splits an equilateral triangle into two congruent triangles. Recall that triangles have side lengths that are in the ratio of. Substitute in the given height into the ratio in order to find the length of the base of the triangle.
The length of the side of the hexagon is twice the length of the base. Substitute in the value of the length of the base to find the side length of the hexagon.
Explanation: This is a regular hexagon; therefore, when we draw in the diagonals, we will create identical equilateral triangles as shown by the figure: Now, the given line segment also serves as the height to the newly created equilateral triangles. This height also splits an equilateral triangle into two congruent triangles. Recall that triangles have side lengths that are in the ratio of. Substitute in the given height into the ratio in order to find the length of the base of the triangle. The length of the side of the hexagon is twice the length of the base.
Substitute in the value of the length of the base to find the side length of the hexagon. If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors. Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org. Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.