How To Find An Apothem

Hello and welcome to the octagon reckoner, the ultimate tool for everything octangular. Here, nosotros hope you will notice an splendid tool for calculating the diagonals, perimeter, circumradius, inradius, and area of a regular octagon. In this octagon surface area calculator, you will as well find the answers to the following questions: what is an octagon, how many sides does an octagon have, how to observe the area of a regular octagon or how to depict an octagon.

We volition 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-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 forth the lines of, "An octagon is a polygon with viii sides delimiting a closed expanse." Anyone with a bones agreement of Greek should be able to comfortably answer the question how many sides does an octagon have without any notions of mathematics. That is because "Octo-" in Greek means eight, then information technology is rubber to presume an octagon has 8 sides or that an octopus has eight legs.

Nosotros will swoop a bit deeper into the Greek origin of the octagon shape when nosotros talk near octagon angles, simply, for at present, let's stick to octagon sides. If someone asks, "how many sides does an octagon have?" the answer is always the same, but if they ask nearly the length of those sides, nosotros cannot give one answer. Every side of an octagon can have unlike lengths and still be an octagon; there are no restrictions on that front. However, when all of an octagon'south internal angles and side lengths are the same, it is chosen a regular octagon and has special backdrop, equally nosotros will see in the post-obit sections.

Octagon angles: What is an octagon?

If nosotros look at the origin of the discussion "Octagon" information technology comes from the Greek meaning "eight angles". The effect of this is a polygon equanimous of eight vertices joined by eight straight lines. The eight sides are a necessary consequence of having viii angles. In fact, the standard octagon definition defines the octagon every bit having eight 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 saying "an eight-sided polygon" as well equally "a polygon with eight angles".

While the sides of an octagon tin can have nearly whatsoever length, the octagon angles are restricted. In that location is not a brake imposed on each bending but on the sum of them. This restriction is geometric since it would be impossible to join all 8 sides together without post-obit this dominion. For whatsoever 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°/eight = 135° (if you adopt other units, feel free to use our angle converter). In our opinion, a regular octagon is much prettier than all other octagons. Afterwards in the text, nosotros'll see how to find the area of an octagon and the tricks associated with a regular octagon.

Surface area of an octagon: How to find the surface area of a regular octagon?

If you want to know how to find the area of a regular octagon past hand, the easiest process is to employ the standard formula for the area of a regular polygon. The formula is the following:

area of regular polygon = perimeter * apothem / two,

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

💡 Nosotros can calculate the perimeter a few means: 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, y'all can also utilise our perimeter of a polygon calculator.

A play tricks to remembering this formula is to understand where it comes from. If you lot look at the apothem, y'all can see that it's the height of a triangle made past taking a line from the vertices of the polygon/octagon to the heart of it. The resulting triangle is what is called an isosceles triangle, and its expanse is:

area of triangle = base of operations * height / two

as we explained in the triangle surface area calculator. Note that the base of the triangle is the length of a side of the octagon. Since there are every bit many of these triangles as the polygon has sides (8 for an octagon), you take to multiply the expanse of this triangle by the number of sides. You volition obtain the full area of the octagon:

area of octagon = 8 * base * summit / two = perimeter * apothem / two.

These tricks work for any polygon, e.g., hexagons and any other polygon you tin remember of, every bit long as information technology is regular. Autonomously from using triangles, in that location are other tricks you can utilise to calculate the area of an octagon if you lot don't remember the formula, only they will not work for other polygons. For example, if you imagine an octagon shape within a foursquare, you tin see that the difference is merely four right triangles. "How to find the expanse of a regular octagon with this information?", you might ask. Well, it is very easy:

Calculate the surface area of the foursquare (the side is 2 * apothem),
Calculate the sides of the right triangles either by using the 45 45 90 triangle estimator or the fact that they are right isosceles triangles, and utilize the special right triangles calculator.
Subtract the area of a right triangle four times from the area of the square.
Enjoy success!

Alternatively, yous tin use this flim-flam. If y'all organize the right triangles correctly, you can construct a square from all 4 of them. In this case, the hypotenuse is also the side of the octagon. Then y'all can summate the expanse of the parallelogram you just made from the four right triangles and subtract it from the area of the big foursquare.

On top of these, you can get fifty-fifty more than creative. For example, imagine that the octagon is composed of a rectangle with two trapezoids, with one in a higher place and below the rectangle. In this case, information technology will be much easier to calculate the area since y'all but have to sum upwardly the expanse 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 xx diagonals; the longest ones are on its axes of symmetry, and they come across at the fundamental point, O, which is too the origin of symmetry. In that location are 3 types of diagonals (take a look at the image to a higher place in the "Surface area of an octagon" section for reference):

Short diagonals, for example, AG, FH, or CE;
Medium diagonals, such every bit AF or Be (as well called the height of an octagon); and
Long diagonals, for instance, AE or BF.

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

Short diagonal s = a * √(2 + √2)
Medium diagonal thousand = a * (1 + √2)
Long diagonal 50 = a * √(4 + 2√2)

If yous're feeling lazy or detect yourself in a blitz, just use the octagon computer.

Circumradius and inradius

Nosotros should also talk most the circumradius and the inradius. Our octagon expanse estimator is capable of finding the radii of the circumscribed and inscribed circles.

You tin can notice that the circumradius is only half of the length of the longest diagonal:

R = l/2 = a / 2 * √(4 + two√2)

Similarly, the inradius is the aforementioned as the apothem, which is merely half of the octagon's elevation:

r = m/2 = a/two * (i + √ii)

Octagon shape: How to draw an octagon?

It might seem easy to draw an octagon at showtime, and, in principle, it is. But depict any shape with viii directly sides, and y'all're done. But typically, what people want to depict is non any octagon, just a regular octagon, as information technology'southward the octagon shape that comes to heed first. So permit's see how to describe an octagon as regularly and as hands every bit possible.

The most precise way would be to use proper drawing tools and start drawing the regular octagon shape ane side at a time, joining them with the corresponding 135° regular octagon angles. But not anybody has such tools laying effectually, so it is not a very applied method. To obtain the octagon shape we are looking for, it is best to get-go with its circumference since it can exist drawn by hand (not recommended) or by using a glass, cup, or even a coin.

How to draw an octagon shape? use your imagination

With the circumference drawn, start dividing the corresponding circle area into halves. First, you will brand 2 one-half-circles. Then, split up them in half over again and get four quarter circles. Halve information technology once more, and you lot finish up with eight circle-eights. Nosotros can now accept the points of division along the circumference and bring together the adjacent ones with a straight line. The result is a perfectly regular octagon, minus the human mistake you may make while cartoon and halving.

Another how-to-describe an octagon trick would be the i we accept already discussed when learning how to observe the area of an octagon with squares. Start by drawing a big square and so 'chop off' its corners. The mathematical term for this process is truncating the square. If yous do it correct, you will go a regular octagon shape out of it. This method is not every bit precise as the previous ane, but it's easier to perform without whatsoever tools at all.

How to apply the octagon calculator

Using the octagon computer isn't too complicated, but only in case someone might accept any doubts and for the sake of completeness, allow'southward become over the features and uses of this octagon expanse calculator. First of all, we should wait at the unlike fields and what they hateful.

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;
Area - Space enclosed past the octagon itself;
Longest diagonal - Length of the line joining 2 furthest vertices;
Medium diagonal - Length of the line joining 2 vertices with 3 vertices between them;
Shortest diagonal - Length of the line joining ii 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 ways, it is time to see how to utilise the octagon calculator to obtain the values you are looking for easily. The all-time feature of this calculator is that it merely requires one input to calculate the rest of the values. This innovation makes the octagon reckoner the fastest way to calculate any properties of an octagon by a significant margin.

Octagons in existent life: the octagon business firm

And so far, nosotros take talked near the octagon definition and how to draw an octagon. We've seen pictures of octagons and even answered the (now obvious) question of how many sides does an octagon accept. Now, it is time to meet how octagons are used in existent life. The octagon shape is easier to manufacture and use in designs than a circumvolve due to the flat sides and the octagon angles. We can often find octagon tiles every bit flooring or in a bathroom. There are even houses that have an octagon shape, well-nigh notably the famous "Octagon Business firm".

The Octagon Firm was built in an octagonal shape in Washington D.C., USA for Colonel John Tayloe III. For this reason, it'southward besides known equally the Colonel John Tayloe 3 House. It follows the American "tradition" of naming buildings later on their shape, as they did with the Pentagon. The Pentagon is the Us Department of Defence headquarters and was built in the shape of a regular pentagon.

The Octagon house has many interesting architectural features, such as a triangular service stairway, a bigger oval stairway, and the obvious octagonal shape of the building. On top of such unique characteristics, several ghost stories surround the business firm, partly because the owner was an important 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 different 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 course of octagon tiles. The shape of a regular octagon means that we can combine octagon tiles and square tiles to fill any room's flooring completely, no affair its foursquare footage.

Using a different pattern in each type of tile or using different colors, these combinations would permit you to create a beautiful flooring for your kitchen or bathroom. Just remember to use our tile estimator to assist you!

Octagon aperture in a camera — Source: Wikimedia

Another heady use of octagons is within photographic cameras. In particular, the shutter protects the sensor from exposure to lite when the camera is non in use. This shutter, or aperture, is always made following a regular polygonal shape, including the octagon shape. Yous can learn more virtually aperture and how to play with it to get more professional-looking pictures with our aperture area figurer.

Before you lot caput out to check that computer, nosotros want to give you a quick tip/trick. When you lot encounter a flick of a bright source of low-cal, you lot might accept noticed that it looks like a star with a dissimilar number of pointy arms, depending on the camera. This observation is the primal to figuring out if your photographic camera has an octagon discontinuity or not: the number of pointy artillery is precisely equal to the number of sides of the polygon that makes up the aperture. And how many sides does an octagon have? Exactly! So you only demand to await for 'star shapes' with 8 pointy arms, and y'all'll know that this picture show was taken using an octagonal aperture. It works every fourth dimension - it's physics, and, more than specifically, a diffraction design.

FAQ

What are the concave and convex octagons?

A convex octagon has all of its interior angles less than 180°. A concave octagon has at to the lowest degree one interior angle greater than 180°. A regular octagon is a convex octagon, as all of its angles are 135°.

What is the sum of the angles of an octagon?

The sum of the angles of an octagon is 1080°. The sum of the interior angles of a polygon with n number of sides is ever (n − 2) × 180°. Thus, an octagon's angles add together up to (eight − 2) × 180° = 1080°.

How do I calculate the surface area of an octagon of side 5 cm?

The formula estimating a regular octagon's surface area is A = 2 × (1+√ii) × a^two, where a is the side length. Thus, your octagon with a side length of v cm has an expanse of 2 × (1+√2) × v² = 120.71 cm² .

What is the maximum side length of a regular octagon made using letter of the alphabet-sized newspaper?

The formula for calculating side length, a, using the diameter, d, is a = d × sin α, where α is the isosceles triangles' apex angle.
The dimension of a standard alphabetic character in the US is 8.5 × eleven inches. So, we can cut a square of maximum length 8.v × 8.five inches. Hence, the octagon'south maximum diameter, d, is eight.five in. Thus, the side length = 8.5 in × sin 45 = 6 in.

Álvaro Díez and Bogna Szyk