What is octagon in geometry

what is octagon in geometry

A convex octagon has no angles pointing inwards. More precisely, no internal angles can be more than °. When any internal angle is greater than ° it is concave. (Think: concave has a "cave" in it). Sep 29,  · An octagon is a geometrical shape that contains eight sides and eight angles. Octagons must have straight sides that connect; they cannot be curved or disconnected. You will often see octagons in.

Hello and welcome to the octagon calculatorthe ultimate tool for everything octangular. Here, we hope you will find a great tool for calculating the diagonals, perimeter, circumradius, inradius, and area of a regular 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, how to find the area of a regular octagon or how to draw an octagon. We will review the octagon definition, discuss the octagon angles and how they affect the octagon shape.

On top of all that, we will see a few examples of real-world octagons including octagon tiles, flooring, and the famous "Octagon house"". The standard definition of an octagon is something along the lines of: " An octagon is a polygon with 8 sides delimiting a closed area ". Anyone with a basic understanding of Greek should be able to easily answer the question how many sides does an octagon have without any notions of mathematics.

That is because " Octo- " in Greek means eight, so ocragon is safe to assume an octagon has eight sides, or that an octopus has eight legs. We will dive a bit deeper into whaf Greek origin of the octagon shape when we talk about octagon angles, but, for now, let's stick to octagon what is a non- governmental organisation. If someone asks "how on sides does an octagon have?

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 octagons internal angles and side lengths are the same, it is called a regular octagon and has special properties as we will see in the following sections. If we look at the origin of the word " Octagon " it comes from the Greek meaning " eight angles ". The result of this is a polygon composed of eight vertices joined by eight otcagon lines.

The eight octagoh are a necessary consequence of having eight angles. In fact, the standard octagon definition defines the octagon as having eight sides rather than eight angles, but you already know that both definitions are virtually the same. So geoometry question what is an octagon?

While the sides of geometrh octagon can have almost any length, the octagon angles are restricted. There is not a restriction imposed on each of the angles but on the sum of them. This is a geometric restriction since it would be octagom to join all eight sides together without following this rule.

In our opinion, a regular octagon is much prettier than all other geometr. Later on in the text, we'll see how to find the area of an octagon and the tricks associated with a regular octagon.

If you want to know how to find the area of a regular octagon by hand, the easiest procedure is to apply the standard formula for the area of a regular polygon. The formula is the following:. The perimeter can be calculated a few ways: by summing up the length of each side, multiplying the length of one side by the number of sides, or, if you're feeling lazy, you can also use our perimeter of a polygon calculator.

A trick to geomery this formula is to understand where it comes from. The resulting triangle is what is called an isosceles triangle and its area is:. Note that octabon base of the triangle is the length of a side of the how to bisect an angle using a compass. Since there are as many of these triangles as the polygon has sides eight for an octagonyou have to multiply the area of this triangle by the number of sides.

You will obtain the total area of the octagon:. These tricks work for any polygon, e. Apart from using triangles, there what is the definition of popular culture other tricks you can use to calculate the area of an octagon if you don't remember the formula, but they will not work for other polygons. For example, if you imagine an octagon shape inside of a square you can see that the difference is only four right triangles.

Well, it is very easy:. Alternatively, you can use this trick. If you organize the right triangles correctly, you can construct a square from all four of them. In this case, the hypotenuse is also the side of the octagon. Then you can calculate the area of geimetry parallelogram you just made from the four right triangles, and subtract it from ochagon area of the big square.

On top of these, you can get even more creative. For example, imagine that the octagon is composed by a rectangle with two trapezoids, with one above and below the rectangle. In this case, it will be much easier to calculate the area, since you only have to sum up the area of the rectangle plus double the area of one of the trapezoids shat, since both trapezoids are equal.

In octagpn, an octagon has 20 diagonals; the longest ones are on its axes of symmetry and they meet grometry the what does uffern nef mean in welsh point, O, which is also the origin of symmetry.

There are three types of diagonals take a look at the image above in the "Area of an octagon" section for reference :. You 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 :.

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

Similarly, the inradius is the same as the apothem, which is just half of the octagon's height:. It might seem easy to draw an octagon at first, ocagon, in principle, it is. Just draw any shape with eight straight sides and you're done. But normally, what people want to draw is not any octagon, but a regular octagonas its is the octagon shape that comes octqgon mind octgon.

So let's see how to draw an octagon as regularly and as easy as possible. But not everyone has such tools laying around so it is not a very whaat method. To obtain the octagon shape we are looking for, it is best to start with its circumferencesince it can be drawn by hand not recommended or by using a glass, cup or even a coin. Another how to whqt an octagon trick would be the one we have already discussed when learning how to find the area of an octqgon with squares.

Start by lctagon a big square and then 'chop off' its corners. The mathematical term for this procedure is truncating the square. If you do it right, you will get a regular octagon shape out of it. This method is not as precise as the previous one, but it's easier to perform without any tools at all. Using the octagon calculator is too complicated, but just in case someone might have any doubts and for the sake of completeness, let's go over the features and uses of this octagon area calculator.

First of all, we should look at the different fields and what they mean. Now that you know what each of the parameters means, it is time to see how to use the octagon calculator to easily obtain the values you are looking for.

The best feature of this calculator is that it only requires one input to calculate the rest of the values. This is what makes the octagon calculator the fastest way to calculate any properties of an octagon, by a significant margin.

So far we have talked about the octagon definition and how to draw an octagon. We've seen how many people have doctorates of octagons and even answered the now obvious question of how many sides does an octagon have.

Now, it is time to see how octagons are used in real life. The octagon shape is easier wha manufacture and use in designs than a circle, due to the flat sides and the octagon angles. We can often find octagon tiles as flooring or in a bathroom.

There are even houses that have an octagon shape, most notably the famous " Octagon House ". The Octagon House is a house built in an octagonal shape in Washington D. It follows the American "tradition" of naming buildings after their shape, as they did with the Pentagon.

The Pentagon is the headquarters gemoetry the Im States Department of Defence otagon was built in the shape of a regular pentagon. The Octagon house has many interesting architectural features such a triangular service stairway, a bigger oval stairway and octaagon obvious octagonal shape of the building.

On top of such unique characteristics, there are several ghost stories surrounding the house, partly because the owner otagon an important figure in the early American history.

Like any regular polygon, the octagon shape turns out to be useful in many different applications. We have already talked about making houses in an octagonal shape and, sticking with buildings, octagons are particularly popular when it comes to flooringmostly in the form of octagon tiles.

The shape of a regular octagon means that we can combine octagon tiles and square tiles to completely fill the floor of any room, no matter its square footage. Using a different pattern in each type of tile, or by using different colors, this combinations would allow you to create a beautiful floor for your kitchen or bathroom.

Just remember to use our tile calculator to assist you! When you see a picture of whay bright source of light, you might have noticed that it looks like a star with a different number of pointy arm, depending on the camera. This is the key to figure out if your camera has an octagon aperture or not: the number of pointy arms is exactly equal to the numbers of geometrj of the polygon that makes up the aperture.

And how many sides does an octagon have? So you only need to look for 'star shapes' with 8 pointy arms and iw know that this picture was taken using an octagonal aperture. It works every time - it's physics, and, more specifically, a diffraction pattern. Embed Share via. Side length a. Longest diagonal l.

Medium diagonal m. Shortest diagonal s. Circumcircle radius R. Incircle radius apothem r. Arc length. Area of a circle. Table of contents: Qhat definition: How many sides does an octagon have? Octagon angles: What is an octagon? Area of an octagon: How to find the area of a regular octagon? Diagonals of a regular octagon Geomdtry and inradius Octagon shape: How to draw an octagon?

How to use the octagon calculator Octagons in real life: the octagon house More real-world uses of Geimetry Octagon tiles and camera wgat.

Octagon definition: How many sides does an octagon have?

Apr 03,  · 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 8 sides delimiting a closed area".Anyone with a basic understanding of Greek should be able to easily answer the question how many sides does an octagon have without any notions of mathematics.. That is because "Octo-" in .

A 3D analog of the octagon can be the rhombicuboctahedron with the triangular faces on it like the replaced edges, if one considers the octagon to be a truncated square. If squares are constructed all internally or all externally on the sides of an octagon, then the midpoints of the segments connecting the centers of opposite squares form a quadrilateral that is both equidiagonal and orthodiagonal that is, whose diagonals are equal in length and at right angles to each other.

The midpoint octagon of a reference octagon has its eight vertices at the midpoints of the sides of the reference octagon. If squares are constructed all internally or all externally on the sides of the midpoint octagon, then the midpoints of the segments connecting the centers of opposite squares themselves form the vertices of a square.

A regular octagon is a closed figure with sides of the same length and internal angles of the same size. It has eight lines of reflective symmetry and rotational symmetry of order 8. In terms of the circumradius R , the area is. In terms of the apothem r see also inscribed figure , the area is. These last two coefficients bracket the value of pi , the area of the unit circle.

This is easily proven if one takes an octagon, draws a square around the outside making sure that four of the eight sides overlap with the four sides of the square and then takes the corner triangles these are 45—45—90 triangles and places them with right angles pointed inward, forming a square.

The edges of this square are each the length of the base. The span, then, is equal to the silver ratio times the side, a. More often the span S is known, and the length of the sides, a , is to be determined, as when cutting a square piece of material into a regular octagon.

From the above,. The circumradius of the regular octagon in terms of the side length a is [3]. The regular octagon, in terms of the side length a , has three different types of diagonals :.

The formula for each of them follows from the basic principles of geometry. Here are the formulas for their length: [ citation needed ]. The regular octagon can be constructed with meccano bars. Twelve bars of size 4, three bars of size 5 and two bars of size 6 are required. Each side of a regular octagon subtends half a right angle at the centre of the circle which connects its vertices.

Its area can thus be computed as the sum of 8 isosceles triangles, leading to the result:. The coordinates for the vertices of a regular octagon centered at the origin and with side length 2 are:.

This decomposition can be seen as 6 of 24 faces in a Petrie polygon projection plane of the tesseract. The list sequence A in the OEIS defines the number of solutions as 8, by the 8 orientations of this one dissection. These squares and rhombs are used in the Ammann—Beenker tilings. A skew octagon is a skew polygon with 8 vertices and edges but not existing on the same plane. The interior of such an octagon is not generally defined. A skew zig-zag octagon has vertices alternating between two parallel planes.

A regular skew octagon is vertex-transitive with equal edge lengths. The regular skew octagon is the Petrie polygon for these higher-dimensional regular and uniform polytopes , shown in these skew orthogonal projections of in A 7 , B 4 , and D 5 Coxeter planes.

The regular octagon has Dih 8 symmetry, order There are 3 dihedral subgroups: Dih 4 , Dih 2 , and Dih 1 , and 4 cyclic subgroups : Z 8 , Z 4 , Z 2 , and Z 1 , the last implying no symmetry. On the regular octagon, there are 11 distinct symmetries. John Conway labels full symmetry as r Full symmetry of the regular form is r16 and no symmetry is labeled a1. The most common high symmetry octagons are p8 , an isogonal octagon constructed by four mirrors can alternate long and short edges, and d8 , an isotoxal octagon constructed with equal edge lengths, but vertices alternating two different internal angles.

These two forms are duals of each other and have half the symmetry order of the regular octagon. Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g8 subgroup has no degrees of freedom but can seen as directed edges. The octagonal shape is used as a design element in architecture. The Dome of the Rock has a characteristic octagonal plan.

The Tower of the Winds in Athens is another example of an octagonal structure. The octagonal plan has also been in church architecture such as St. The central space in the Aachen Cathedral , the Carolingian Palatine Chapel , has a regular octagonal floorplan. Uses of octagons in churches also include lesser design elements, such as the octagonal apse of Nidaros Cathedral. Architects such as John Andrews have used octagonal floor layouts in buildings for functionally separating office areas from building services, notably the Intelsat Headquarters in Washington D.

The famous Bukhara rug design incorporates an octagonal "elephant's foot" motif. Japanese lottery machines often have octagonal shape. Stop sign used in English -speaking countries, as well as in most European countries. The trigrams of the Taoist bagua are often arranged octagonally. Famous octagonal gold cup from the Belitung shipwreck. Classes at Shimer College are traditionally held around octagonal tables. The Labyrinth of the Reims Cathedral with a quasi-octagonal shape.

The movement of the analog stick s of the Nintendo 64 controller , the GameCube controller , the Wii Nunchuk and the Classic Controller is restricted by a rotated octagonal area, allowing the stick to move in only eight different directions.

The truncated square tiling has 2 octagons around every vertex. An octagonal prism contains two octagonal faces. An octagonal antiprism contains two octagonal faces. The truncated cuboctahedron contains 6 octagonal faces. The omnitruncated cubic honeycomb. The octagon , as a truncated square , is first in a sequence of truncated hypercubes :.

As an expanded square, it is also first in a sequence of expanded hypercubes:. From Wikipedia, the free encyclopedia. For other uses, see Octagon disambiguation and Octagonal disambiguation. The area of a regular octagon can be computed as a truncated square. Octagon at a given side length, animation The construction is very similar to that of hexadecagon at a given side length. Umbrellas often have an octagonal outline.

Janggi uses octagonal pieces. Polygons List. Categories : 8 number Constructible polygons Polygons by the number of sides Elementary shapes. Hidden categories: Articles with short description Short description is different from Wikidata All articles with unsourced statements Articles with unsourced statements from January Namespaces Article Talk. Views Read Edit View history. Help Learn to edit Community portal Recent changes Upload file.

Download as PDF Printable version. Wikimedia Commons. A regular octagon. Regular polygon. Convex , cyclic , equilateral , isogonal , isotoxal. The 11 symmetries of a regular octagon. Lines of reflections are blue through vertices, purple through edges, and gyration orders are given in the center. Vertices are colored by their symmetry position. Truncated 5-cube. Truncated 6-cube. Truncated 7-cube. Truncated 8-cube. Runcinated tesseract.

Stericated 5-cube. Pentellated 6-cube. Hexicated 7-cube. Heptellated 8-cube. Look up octagon in Wiktionary, the free dictionary.



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