Hello and welcome to the octagon calculator, the ultimate tool for every little thing octangular. Here, us hope girlfriend will uncover a an excellent tool because that calculating the diagonals, perimeter, circumradius, inradius, and area the a consistent octagon. In this octagon area calculator, girlfriend will also find the answers to the following questions: what is an octagon, how countless sides go an octagon have, how to find the area of a continual octagon or just how to attract an octagon.

You are watching: How to find the area of a regular octagon

We will evaluation the octagon definition, comment on the octagon angles and how they affect the octagon shape. On optimal of all that, us will check out a few examples of real-world octagons including octagon tiles, flooring, and the renowned "Octagon house"".


Octagon definition: How many sides go an octagon have?


The standard an interpretation of an octagon is something along the currently of: "An octagon is a polygon through 8 sides delimiting a closed area". Anyone v a basic understanding that Greek should have the ability to easily prize the question how plenty of sides walk an octagon have without any kind of notions of mathematics. The is due to the fact that "Octo-" in Greek way eight, so the is safe to assume an octagon has actually eight sides, or the an octopus has eight legs.

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We will certainly dive a bit deeper right into the Greek beginning of the octagon shape as soon as we talk about octagon angles, but, for now, let"s stick come octagon sides. If who asks "how countless sides does an octagon have?" the prize is always the same, but if castle ask about the length of those sides, there is no one price we deserve to give. Every next of one octagon deserve to have different lengths and also still it is in an octagon; there are no limitations on that front. However, when all of an octagons internal angles and also side lengths room the same, the is called a regular octagon and has one-of-a-kind properties together we will check out in the adhering to sections.


Octagon angles: What is one octagon?


If we look at the origin of words "Octagon" it comes from the Greek an interpretation "eight angles". The an outcome of this is a polygon composed the eight vertices joined by eight right lines. The eight sides are a necessary consequence of having actually eight angles. In fact, the conventional octagon definition defines the octagon as having eight sides quite than eight angles, however you currently know the both interpretations are practically the same. For this reason the concern what is one octagon? is equally well answered by saying "an eight-sided polygon" and "a polygon through eight angles".

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While the sides of one octagon deserve to have practically any length, the octagon angles room restricted. There is no a restriction implemented on every of the angles however on the sum of them. This is a geometric restriction since it would be difficult to sign up with all eight sides together without following this rule. For any type of octagons, the amount of every one of its interior angles is always 1080°.

This means that a continual octagon will have actually internal octagon angle of precisely 1080°/8 = 135° (if friend prefer other units, feel totally free to use our angle converter). In our opinion, a continual octagon is much prettier than all other octagons. After that in the text, we"ll see how to discover the area of an octagon and also the tricks associated with a regular octagon.


Area of an octagon: how to discover the area of a constant octagon?


If you want to know just how to discover the area of a constant octagon by hand, the simplest procedure is to apply the standard formula because that the area that a consistent polygon. The formula is the following:

area of continuous polygon = perimeter * apothem / 2,

where the apothem is the distance for the center of the polygon to the mid-point that a side. The perimeter have the right to be calculation a few ways: by summing increase the length of every side, multiplying the length of one next by the number of sides, or, if you"re feeling lazy, friend can additionally use our perimeter the a polygon calculator.

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A trick to remember this formula is to understand where it come from. If girlfriend look in ~ the apothem you deserve to see the it"s the height of a triangle do by taking a heat from the vertices of the polygon/octagon come the center of it. The resulting triangle is what is referred to as an isosceles triangle and its area is:

area of triangle = basic * elevation / 2

as we described in the triangle area calculator. Note that the basic of the triangle is the size of a side of the octagon. Due to the fact that there are as countless of these triangles together the polygon has sides (eight because that an octagon), you need to multiply the area that this triangle by the variety of sides. You will acquire the complete area that the octagon:

area that octagon = 8 * base * height / 2 = perimeter * apothem / 2.

These tricks work-related for any kind of polygon, e.g., hexagons and also any other polygon you have the right to think of, as lengthy as it is regular. Personal from utilizing triangles, over there are various other tricks you have the right to use to calculate the area of an octagon if friend don"t remember the formula, yet they will certainly not work-related for various other polygons. For example, if you imagine one octagon shape inside that a square you deserve to see that the distinction is only 4 right triangles. "How to discover the area of a consistent octagon with this information?" you could ask. Well, that is an extremely easy:

Enjoy success!
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Alternatively, you deserve to use this trick. If friend organize the right triangles correctly, you deserve to construct a square native all 4 of them. In this case, the hypotenuse is likewise the side of the octagon. Then you can calculate the area of the parallelogram you simply made native the four right triangles, and also subtract it from the area that the large square.

On optimal of these, friend can get even much more creative. Because that example, imagine the the octagon is created by a rectangle through two trapezoids, v one over and listed below the rectangle. In this case, it will be much easier to calculation the area, because you only need to sum increase the area the the rectangle plus double the area of one of the trapezoids, since both trapezoids are equal.


In total, an octagon has 20 diagonals; the longest ones space on its axes the symmetry and they fulfill at the main point, O, i beg your pardon is additionally the origin of symmetry. There are three types of diagonals (take a look at the image over in the "Area of one octagon" ar for reference):

Short diagonals, because that example, AG, FH or CE;Medium diagonals, such together AF or it is in (also dubbed the elevation of one octagon);Long diagonals, for example, AE or BF.

You deserve to derive the formula for each that them with ease using the simple principles that geometry. Right here are the formulas for the length of the diagonals:

Short diagonal line s = a * √(2 + √2)Medium diagonal line m = a * (1 + √2)Long diagonal l = a * √(4 + 2√2).

If you"re emotion lazy or uncover yourself in a rush, simply use the octagon calculator.


We should likewise talk about the circumradius and also the inradius. Our octagon area calculator is capable of finding the radii that the circumscribed and also inscribed circles.

You can notice that the circumradius is simply fifty percent of the size of the longest diagonal:

R = l/2 = a / 2 * √(4 + 2√2)

Similarly, the inradius is the very same as the apothem, i m sorry is just half of the octagon"s height:

r = m/2 = a/2 * (1 + √2)


It could seem simple to attract an octagon in ~ first, and, in principle, that is. Simply draw any type of shape with eight right sides and also you"re done. However normally, what world want to attract is no any octagon, but a regular octagon, together its is the octagon form that pertains to mind first. Therefore let"s see how to attract an octagon together regularly and also as easy as possible.

The most precise way would be to use ideal drawing tools and also start illustration the continual octagon form one next at a time, involvement them with the corresponding 135° continuous octagon angles. But not everyone has such tools laying approximately so that is not a an extremely practical method. To obtain the octagon shape we are looking for, it is best to start with the circumference, because it deserve to be drawn by hand (not recommended) or by utilizing a glass, cup or also a coin.

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With the one drawn, start splitting the matching (calc:752) into halves. First, you will make two half-circles. Then, divide them in half again and also get 4 quarter circles. Halve it once more and you end up with **eight circle-eights**. We can now take it the point out of division along the circumference and join the surrounding ones through a directly line. The result is a perfectly continuous octagon, minus the human error you might make whilst drawing and halving.Another exactly how to draw an octagon trick would be the one we have currently discussed as soon as learning how to discover the area of an octagon with squares. Start by drawing a large square and then "chop off" its corners. The math term because that this procedure is truncating the square. If you do it right, girlfriend will gain a regular octagon shape out that it. This technique is not as an accurate as the ahead one, but it"s easier to do without any kind of tools in ~ all.


Using the octagon calculator is too complicated, but just in instance someone could have any type of doubts and for the services of completeness, let"s walk over the features and uses of this octagon area calculator. Very first of all, we have to look in ~ the different fields and what lock mean.

Side length - that is the size of each side of the continual octagon.Perimeter - amount of the length of all the political parties of the octagonArea - an are enclosed by the octagon itselfLongest diagonal - length of the heat joining 2 furthest verticesMedium diagonal - length of the heat joining 2 vertices through 3 vertices between themShortest diagonal - length of the heat joining 2 vertices through 2 vertices in in between themCircumcircle radius - Radius of the circumference that consists of all 8 vertices that the regular octagonIncircle radius - Radius the the smallest circumference that is tangent to every 8 sides

Now the you understand what each of the parameters means, the is time to see how to use the octagon calculator to easily attain the worths you room looking for. The best feature that this calculator is the it just requires one input to calculation the remainder of the values. This is what makes the octagon calculator the fastest means to calculate any type of properties of an octagon, by a significant margin.


So far we have actually talked around the octagon definition and exactly how to attract an octagon. We"ve seen photos of octagons and even answer the (now obvious) question of how plenty of sides walk an octagon have. Now, that is time come see how octagons are supplied in real life. The octagon form is much easier to manufacture and use in designs 보다 a circle, due to the level sides and also the octagon angles. We deserve to often discover octagon tiles together flooring or in a bathroom. There space even dwellings that have an octagon shape, most notably the famed "Octagon House".

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The Octagon house is a house built in an octagonal shape in Washington D.C. In the USA. It was developed for Colonel john Tayloe III. This is the reason, it"s likewise known as Colonel man Tayloe III House. It follows the American "tradition" of naming buildings after your shape, together they did through the Pentagon. The Pentagon is the headquarters that the United claims Department that Defence and was constructed in the shape of a continual pentagon.

The Octagon home has plenty of interesting architectural attributes such a triangular company stairway, a larger oval stairway and also the noticeable octagonal form of the building. On peak of such distinctive characteristics, over there are number of ghost stories surrounding the house, partly because the owner was crucial figure in the early American history.


Like any type of regular polygon, the octagon form turns out to be valuable in countless different applications. We have currently talked around making homes in an octagonal shape and, sticking with buildings, octagons are particularly popular once it comes to flooring, largely in the kind of octagon tiles. The shape of a continual octagon method that we can integrate octagon tiles and square tiles to completely fill the floor of any room, no issue its square footage. Utilizing a various pattern in each form of tile, or by using different colors, this combine would allow you to develop a beautiful floor for your kitchen or bathroom. Just remember to usage our tile calculator to assist you!

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Another interesting use that octagons is inside photographic cameras. In particular, the shutter the protects the sensor native exposure come light when the camera is not in use. This shutter, or aperture, is always made adhering to a constant polygonal shape, consisting of the octagon shape. You deserve to learn an ext about aperture and also how to play v it come get more professional looking pictures with our (calc:1027).

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Before friend head out to inspect that calculator we want to offer you a fast tip/trick. When you watch a snapshot of a bright source of light, you could have noticed the it looks favor a star v a different variety of pointy arm, depending on the camera. This is the an essential to figure out if her camera has an octagon aperture or not: the variety of pointy eight is exactly equal come the numbers of political parties of the polygon that renders up the aperture. And how countless sides go an octagon have? Exactly! So friend only should look for "star shapes" through 8 pointy arms and also you"ll recognize that this photo was taken utilizing an octagonal aperture. It functions every time - it"s physics, and, more specifically, a diffraction pattern.