How To Find The Diameter Of An Octagon

Hullo and welcome to the octagon calculator, the ultimate tool for everything octangular. Here, we hope you will find an excellent tool for calculating the diagonals, perimeter, circumradius, inradius, and area of a regular octagon. In this octagon area reckoner, you will also find the answers to the following questions: what is an octagon, how many sides does an octagon have, how to find the surface area of a regular octagon or how to draw an octagon.

We will review the octagon definition, discuss the octagon angles and how they touch on the octagon shape. On top of all that, nosotros volition come across a few examples of real-earth octagons including octagon tiles, flooring, and the famous "Octagon house".

Octagon definition: How many sides does an octagon have?

The standard definition of an octagon is something along the lines of, "An octagon is a polygon with viii sides delimiting a airtight area." Anyone with a basic understanding of Greek should exist able to comfortably answer the question how many sides does an octagon accept without any notions of mathematics. That is because "Octo-" in Greek means eight, then it is safe to presume an octagon has eight sides or that an octopus has viii legs.

Nosotros volition dive a bit deeper into the Greek origin of the octagon shape when we talk about octagon angles, but, for now, let's stick to octagon sides. If someone asks, "how many sides does an octagon accept?" the answer is ever the same, just if they ask nearly the length of those sides, nosotros cannot requite one reply. Every side of an octagon can have different lengths and still be an octagon; there are no restrictions on that front. However, when all of an octagon's internal angles and side lengths are the same, information technology is chosen a regular octagon and has special properties, as we volition see in the following sections.

Octagon angles: What is an octagon?

If nosotros look at the origin of the give-and-take "Octagon" it comes from the Greek meaning "viii angles". The result of this is a polygon composed of eight vertices joined by eight straight lines. The eight sides are a necessary consequence of having eight angles. In fact, the standard octagon definition defines the octagon as having viii sides rather than eight angles, but you already know that both definitions are virtually the same. Then the question what is an octagon? is equally well answered by maxim "an 8-sided polygon" as well every bit "a polygon with 8 angles".

While the sides of an octagon can have nigh whatsoever length, the octagon angles are restricted. At that place is not a brake imposed on each angle but on the sum of them. This restriction is geometric since information technology would be impossible to join all 8 sides together without following this rule. For whatever octagons, the sum of all of its internal angles is always 1080°.

That means that a regular octagon will have internal octagon angles of exactly 1080°/8 = 135° (if you prefer other units, feel free to use our bending converter). In our opinion, a regular octagon is much prettier than all other octagons. Later on on in the text, nosotros'll see how to find the expanse of an octagon and the tricks associated with a regular octagon.

Area of an octagon: How to observe the surface area of a regular octagon?

If you want to know how to find the area of a regular octagon by manus, the easiest procedure is to apply the standard formula for the surface area of a regular polygon. The formula is the following:

area of regular polygon = perimeter * apothem / ii,

where the apothem is the distance from the heart of the polygon to the mid-point of a side.

💡 Nosotros tin calculate the perimeter a few means: past summing upward the length of each side, multiplying the length of one side by the number of sides, or, if yous're feeling lazy, yous can also utilize our perimeter of a polygon estimator.

A play tricks to remembering this formula is to understand where it comes from. If you expect at the apothem, you can come across that it's the height of a triangle made by taking a line from the vertices of the polygon/octagon to the center of it. The resulting triangle is what is called an isosceles triangle, and its surface area is:

expanse of triangle = base * height / 2

as we explained in the triangle area calculator. Note that the base of operations of the triangle is the length of a side of the octagon. Since there are every bit many of these triangles equally the polygon has sides (viii for an octagon), you accept to multiply the surface area of this triangle past the number of sides. You lot volition obtain the total area of the octagon:

expanse of octagon = viii * base * height / 2 = perimeter * apothem / 2.

These tricks work for whatsoever polygon, eastward.grand., hexagons and any other polygon you tin can think of, every bit long equally information technology is regular. Apart from using triangles, at that place are other tricks you can use to calculate the area of an octagon if you don't call back the formula, but they will not work for other polygons. For example, if you imagine an octagon shape inside a square, you tin can see that the difference is only iv correct triangles. "How to discover the expanse of a regular octagon with this information?", yous might inquire. Well, it is very like shooting fish in a barrel:

Calculate the expanse of the square (the side is two * apothem),
Calculate the sides of the right triangles either by using the 45 45 xc triangle calculator or the fact that they are right isosceles triangles, and use the special correct triangles reckoner.
Subtract the area of a correct triangle 4 times from the area of the square.
Savor success!

Alternatively, y'all can use this trick. If yous organize the right triangles correctly, yous can construct a square from all four of them. In this example, the hypotenuse is also the side of the octagon. So you can calculate the area of the parallelogram you only made from the four right triangles and subtract it from the expanse of the big foursquare.

On peak of these, yous can get fifty-fifty more creative. For instance, imagine that the octagon is composed of a rectangle with two trapezoids, with one in a higher place and below the rectangle. In this example, it volition exist much easier to calculate the area since you lot only have to sum upwardly the area of the rectangle plus double the area of ane of the trapezoids, since both trapezoids are equal.

Diagonals of a regular octagon

In total, an octagon has twenty diagonals; the longest ones are on its axes of symmetry, and they meet at the central betoken, O, which is also the origin of symmetry. There are iii types of diagonals (take a wait at the paradigm above in the "Area of an octagon" department for reference):

Brusk diagonals, for example, AG, FH, or CE;
Medium diagonals, such as AF or BE (besides chosen the top of an octagon); and
Long diagonals, for example, AE or BF.

You lot can derive the formula for each of them with ease using the basic principles of geometry. Here are the formulas for the length of the diagonals:

Short diagonal south = a * √(ii + √2)
Medium diagonal k = a * (1 + √2)
Long diagonal l = a * √(4 + ii√2)

If you're feeling lazy or discover yourself in a blitz, just utilize the octagon estimator.

Circumradius and inradius

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

Yous can find that the circumradius is simply one-half of the length of the longest diagonal:

R = fifty/two = a / 2 * √(4 + 2√2)

Similarly, the inradius is the same equally the apothem, which is just half of the octagon'south tiptop:

r = yard/two = a/2 * (i + √2)

Octagon shape: How to draw an octagon?

It might seem easy to describe an octagon at start, and, in principle, information technology is. Just describe any shape with viii straight sides, and you're done. But typically, what people desire to draw is not whatsoever octagon, merely a regular octagon, as it'due south the octagon shape that comes to mind first. So let'southward see how to draw an octagon as regularly and as easily equally possible.

The most precise manner would be to use proper drawing tools and start drawing the regular octagon shape one side at a fourth dimension, joining them with the corresponding 135° regular octagon angles. But not everyone has such tools laying around, then it is not a very applied method. To obtain the octagon shape nosotros are looking for, it is best to start with its circumference since it tin can exist fatigued past hand (not recommended) or by using a drinking glass, cup, or even a coin.

How to draw an octagon shape? use your imagination

With the circumference fatigued, kickoff dividing the respective circle area into halves. First, you will brand 2 half-circles. Then, split them in half once more and get four quarter circles. Halve it in one case more, and you end up with viii circle-eights. We can now take the points of division forth the circumference and join the adjacent ones with a straight line. The outcome is a perfectly regular octagon, minus the homo error yous may make while drawing and halving.

Another how-to-depict an octagon trick would exist the 1 we have already discussed when learning how to detect the area of an octagon with squares. Start past drawing a large square and and then 'chop off' its corners. The mathematical term for this procedure is truncating the square. If y'all do it right, you will become a regular octagon shape out of information technology. This method is non as precise as the previous 1, but it's easier to perform without whatever tools at all.

How to use the octagon calculator

Using the octagon estimator isn't too complicated, but merely in example someone might have any doubts and for the sake of abyss, let's get over the features and uses of this octagon area calculator. Start of all, we should look at the dissimilar fields and what they mean.

Side Length - It is the length of each side of the regular octagon;
Perimeter - Sum of the length of all the sides of the octagon;
Expanse - Infinite enclosed by the octagon itself;
Longest diagonal - Length of the line joining ii furthest vertices;
Medium diagonal - Length of the line joining 2 vertices with three vertices betwixt them;
Shortest diagonal - Length of the line joining two vertices with 2 vertices in between them;
Circumcircle radius - Radius of the circumference that contains all 8 vertices of the regular octagon; and
Incircle radius - Radius of the smallest circumference that is tangent to all 8 sides.

Now that you know what each parameter means, it is time to run into how to use the octagon calculator to obtain the values you lot are looking for easily. The best characteristic of this calculator is that it just requires i input to calculate the remainder of the values. This innovation makes the octagon calculator the fastest way to calculate any properties of an octagon by a significant margin.

Octagons in real life: the octagon house

So far, we have talked about the octagon definition and how to describe an octagon. Nosotros've seen pictures of octagons and even answered the (now obvious) question of how many sides does an octagon have. At present, information technology is time to see how octagons are used in real life. The octagon shape is easier to industry and utilize in designs than a circle due to the flat sides and the octagon angles. We can often find octagon tiles equally floor or in a bath. There are even houses that have an octagon shape, most notably the famous "Octagon House".

The Octagon House was built in an octagonal shape in Washington D.C., USA for Colonel John Tayloe III. For this reason, it'due south as well known as the Colonel John Tayloe III House. It follows the American "tradition" of naming buildings after their shape, as they did with the Pentagon. The Pentagon is the United States Section of Defence headquarters and was built in the shape of a regular pentagon.

The Octagon business firm has many interesting architectural features, such every bit a triangular service stairway, a bigger oval stairway, and the obvious octagonal shape of the building. On acme of such unique characteristics, several ghost stories surround the firm, partly because the owner was an of import figure in early American history.

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

Like any regular polygon, the octagon shape turns out to be useful in many unlike applications. We have already talked about making houses in an octagonal shape, and, sticking with buildings, octagons are specially popular when it comes to flooring, mainly in the form of octagon tiles. The shape of a regular octagon means that nosotros can combine octagon tiles and foursquare tiles to fill whatsoever room's floor completely, no matter its square footage.

Using a dissimilar pattern in each type of tile or using unlike colors, these combinations would allow you to create a beautiful flooring for your kitchen or bathroom. Just remember to use our tile calculator to aid y'all!

Octagon aperture in a camera — Source: Wikimedia

Some other exciting utilise of octagons is inside photographic cameras. In detail, the shutter protects the sensor from exposure to light when the camera is not in use. This shutter, or aperture, is always made following a regular polygonal shape, including the octagon shape. You can learn more about aperture and how to play with information technology to get more than professional-looking pictures with our aperture area calculator.

Before you head out to bank check that computer, we desire to give you a quick tip/pull a fast one on. When you lot see a film of a brilliant source of light, you might have noticed that it looks like a star with a dissimilar number of pointy arms, depending on the camera. This observation is the key to figuring out if your camera has an octagon aperture or not: the number of pointy arms is precisely equal to the number of sides of the polygon that makes upward the aperture. And how many sides does an octagon take? Exactly! Then you merely need to look for 'star shapes' with viii pointy artillery, and yous'll know that this motion picture was taken using an octagonal discontinuity. Information technology works every time - it's physics, and, more specifically, a diffraction pattern.

Álvaro Díez and Bogna Szyk