Hello and welcome to the octagon calculator, the ultimate tool for whatever octangular. Here, we hope girlfriend will uncover a an excellent tool for calculating the diagonals, perimeter, circumradius, inradius, and area the a consistent octagon. In this octagon area calculator, you will also find the answers to the following questions: what is an octagon, how many sides does an octagon have, just how to find the area of a regular octagon or just how to attract an octagon.

You are watching: What is the area of a regular octagon with a side of 8

We will testimonial the octagon definition, talk about the octagon angles and also how they influence the octagon shape. On peak of every that, we will check out a couple of examples of real-world octagons consisting of octagon tiles, flooring, and the famed "Octagon house"".

Octagon definition: How plenty of sides go an octagon have?

The standard an interpretation of an octagon is something follow me the present of: "An octagon is a polygon through 8 sides delimiting a close up door area". Anyone v a simple understanding that Greek should be able to easily answer the concern how plenty of sides go an octagon have without any kind of notions of mathematics. The is because "Octo-" in Greek means eight, so that is for sure to i think an octagon has eight sides, or that an octopus has eight legs.


We will dive a little bit deeper right into the Greek origin of the octagon shape once we talk around octagon angles, but, because that now, let's stick to octagon sides. If someone asks "how plenty of sides does an octagon have?" the price is always the same, however if they ask about the length of those sides, over there is not one price we have the right to give. Every side of one octagon deserve to have different lengths and still be an octagon; there room no limitations on that front. However, when all of an octagons inner angles and also side lengths space the same, it is called a regular octagon and has distinct properties together we will check out in the complying with sections.

Octagon angles: What is an octagon?

If we look in ~ the beginning of words "Octagon" it comes from the Greek meaning "eight angles". The an outcome of this is a polygon composed of eight vertices join by eight straight lines. The eight sides are a necessary an effect of having eight angles. In fact, the standard octagon an interpretation defines the octagon as having eight sides rather than eight angles, however you currently know the both meanings are basically the same. Therefore the concern what is an octagon? is equally well answered by saying "an eight-sided polygon" and also "a polygon with eight angles".

While the sides of one octagon deserve to have almost any length, the octagon angles room restricted. Over there is not a restriction enforced on each of the angles yet on the sum of them. This is a geometric restriction due to the fact that it would certainly be impossible to join all eight sides with each other without complying with this rule. For any type of octagons, the amount of every one of its interior angles is constantly 1080°.

This means that a constant octagon will have internal octagon angles of specifically 1080°/8 = 135° (if you prefer various other units, feel free to use our angle converter). In ours opinion, a consistent octagon is much prettier 보다 all various other octagons. Afterwards in the text, we'll see exactly how to find the area of an octagon and the tricks connected with a continuous octagon.

Area of an octagon: exactly how to uncover the area the a continuous octagon?

If you want to know just how to uncover the area the a constant octagon by hand, the simplest procedure is to apply the standard formula for the area that a regular polygon. The formula is the following:

area of continual polygon = perimeter * apothem / 2,

where the apothem is the distance for the facility of the polygon to the mid-point that a side. The perimeter have the right to be calculated a few ways: by summing up the length of each side, multiply the length of one side by the number of sides, or, if you're feeling lazy, girlfriend can also use our perimeter that a polygon calculator.


A trick come remember this formula is to understand where it comes from. If you look at the apothem you deserve to see that it's the height of a triangle made by acquisition a heat from the vertices of the polygon/octagon to the center of it. The result triangle is what is called an isosceles triangle and also its area is:

area the triangle = basic * height / 2

as we described in the triangle area calculator. Note that the base of the triangle is the size of a side of the octagon. Because there space as many of these triangles together the polygon has sides (eight because that an octagon), you have to multiply the area the this triangle by the variety of sides. Girlfriend will obtain the complete area the the octagon:

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

These tricks occupational for any kind of polygon, e.g., hexagons and also any other polygon you deserve to think of, as lengthy as that is regular. Apart from making use of triangles, there are various other tricks you deserve to use to calculate the area of an octagon if you don't mental the formula, however they will not work for other polygons. Because that example, if friend imagine one octagon shape inside of a square you have the right to see the the difference is only 4 right triangles. "How to find the area that a constant octagon through this information?" you could ask. Well, the is really easy:

Enjoy success!

Alternatively, you can use this trick. If girlfriend organize the ideal triangles correctly, you can construct a square native all 4 of them. In this case, the hypotenuse is likewise the side of the octagon. Then you deserve to calculate the area that the parallel you just made indigenous the 4 right triangles, and subtract the from the area of the big square.

On optimal of these, girlfriend can acquire even much more creative. For example, imagine the the octagon is composed by a rectangle v two trapezoids, through one above and listed below the rectangle. In this case, it will certainly be much simpler to calculate the area, because you only have to sum increase the area of the rectangle plus double the area of one of the trapezoids, because both trapezoids space equal.

Diagonals that a continuous octagon

In total, an octagon has actually 20 diagonals; the longest ones space on the axes that symmetry and they fulfill at the central point, O, i m sorry is also the origin of symmetry. There space three varieties of diagonals (take a look in ~ the image over in the "Area of one octagon" ar for reference):

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

You have the right to derive the formula for each of them with ease utilizing the straightforward principles of geometry. Below are the formulas for the length of the diagonals:

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

If you're feeling lazy or uncover yourself in a rush, just use the octagon calculator.

Circumradius and also inradius

We should additionally talk about the circumradius and also the inradius. Ours octagon area calculator is qualified of finding the radii of the circumscribed and inscribed circles.

You can notification that the circumradius is simply half of the length of the longest diagonal:

R = l/2 = a / 2 * √(4 + 2√2)

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

r = m/2 = a/2 * (1 + √2)

Octagon shape: just how to attract an octagon?

It can seem basic to draw an octagon in ~ first, and, in principle, the is. Simply draw any kind of shape through eight straight sides and you're done. However normally, what human being want to draw is not any octagon, however a regular octagon, as its is the octagon form that pertains to mind first. Therefore let's see just how to draw an octagon as regularly and as basic as possible.

The many precise method would it is in to use ideal drawing tools and start illustration the continuous octagon form one next at a time, authorized them through the matching 135° continuous octagon angles. But not everyone has such tools laying roughly so that is not a an extremely practical method. To obtain the octagon shape we space looking for, it is finest to start with the circumference, due to the fact that it deserve to be attracted by hand (not recommended) or by making use of a glass, cup or even a coin.

With the circumference drawn, start dividing the matching circle area right into halves. First, you will certainly make two half-circles. Then, division them in fifty percent again and also get 4 quarter circles. Halve the once more and you end up with **eight circle-eights**. We have the right to now take it the point out of division along the circumference and also join the surrounding ones v a directly line. The result is a perfectly continual octagon, minus the person error you might make whilst drawing and also halving.Another how to attract an octagon trick would certainly be the one us have already discussed when learning how to uncover the area of an octagon with squares. Start by drawing a big square and also then 'chop off' the corners. The math term for this procedure is truncating the square. If you carry out it right, friend will acquire a continual octagon shape out of it. This technique is not as an exact as the previous one, but it's less complicated to perform without any tools in ~ all.

How to usage the octagon calculator

Using the octagon calculator is also complicated, however just in case someone could have any kind of doubts and also for the benefits of completeness, let's go over the features and uses that this octagon area calculator. Very first of all, we should look at the different fields and also what lock mean.

Side length - that is the length of every side the the continual octagon.Perimeter - sum of the size of every the political parties of the octagonArea - space enclosed by the octagon itselfLongest diagonal line - size of the line joining 2 the furthest verticesMedium diagonal - length of the heat joining 2 vertices v 3 vertices between themShortest diagonal - size of the line joining 2 vertices v 2 vertices in between themCircumcircle radius - Radius of the one that includes all 8 vertices that the consistent octagonIncircle radius - Radius of the smallest circumference the is tangent to every 8 sides

Now the you know what every of the parameters means, it is time to see just how to use the octagon calculator to easily obtain the values you room looking for. The ideal feature the this calculator is that it just requires one input to calculate the remainder of the values. This is what renders the octagon calculator the fastest way to calculate any kind of properties of an octagon, through a far-ranging margin.

Octagons in real life: the octagon house

So much we have talked around the octagon an interpretation and exactly how to draw an octagon. We've seen photos of octagons and even answer the (now obvious) inquiry of how plenty of sides does an octagon have. Now, that is time to see just how octagons are used in real life. The octagon shape is less complicated to manufacture and use in designs 보다 a circle, because of the flat sides and the octagon angles. We have the right to often discover octagon tiles together flooring or in a bathroom. There room even houses that have actually an octagon shape, many notably the famed "Octagon House".

The Octagon house is a house constructed in an octagonal form in Washington D.C. In the USA. That was built for Colonel john Tayloe III. This is the reason, it's additionally known together Colonel john Tayloe III House. It complies with the American "tradition" of naming buildings after their shape, as they did through the Pentagon. The Pentagon is the headquarters that the United states Department that Defence and also was developed in the shape of a continual pentagon.

The Octagon house has plenty of interesting architectural features such a triangular company stairway, a larger oval stairway and the evident octagonal shape of the building. On peak of such unique characteristics, there are number of ghost stories neighboring the house, partly because the owner was an important figure in the early American history.

More real-world provides of Octagons: Octagon tiles and also camera apertures

Like any type of regular polygon, the octagon form turns out to be useful in countless different applications. Us have currently talked around making homes in an octagonal shape and, sticking through buildings, octagons are particularly popular once it pertains to flooring, mostly in the kind of octagon tiles. The form of a constant octagon means that us can integrate octagon tiles and square tiles to fully fill the floor of any type of room, no matter its square footage. Making use of a various pattern in each form of tile, or through using various colors, this combinations would enable you to create a beautiful floor for your kitchen or bathroom. Simply remember to use our brick calculator to help you!

Another exciting use that octagons is within photographic cameras. In particular, the shutter that protects the sensor native exposure come light when the camera is not in use. This shutter, or aperture, is always made adhering to a regular polygonal shape, including the octagon shape. You deserve to learn much more about aperture and how to play v it come get much more professional looking photos with ours aperture area calculator.

See more: Can A 38 Special Shoot 357 Magnum Can Shoot 38 Special Shoot 357 Rounds?

Before you head out to inspect that calculator we desire to provide you a fast tip/trick. When you see a picture of a bright source of light, you can have noticed that it looks like a star with a different number of pointy arm, depending upon the camera. This is the vital to number out if her camera has an octagon aperture or not: the number of pointy eight is exactly equal to the number of sides of the polygon that provides up the aperture. And how many sides go an octagon have? Exactly! So girlfriend only should look because that 'star shapes' v 8 pointy arms and you'll know that this picture was taken using an octagonal aperture. It functions every time - it's physics, and, an ext specifically, a diffraction pattern.