Area Of An Octagon Formula

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Area of Octagon

The area of octagon is defined as the total amount of area that is enclosed by all octa(eight) sides of the octagon. Octagon is an eight-sided polygon. The area of the octagon can be calculated by diving it into 8 equal isosceles triangles. In this mini-lesson, we will discuss the area of the octagon in detail.

What Is the Area of Octagon?

We already read that octagon is a 2-dimensional shape with 8 sides. The area of the octagon is the area that is within the eight sides of the octagon. To calculate the area of the octagon we can use the area of an isosceles triangle. We divide the area of the octagon into eight equal isosceles triangles and calculate it accordingly.

The general formula we use to calculate the area of the octagon is:

Area of Octagon Formula

2s2(1+√2), where s is the length of the side of the octagon.

How to Calculate Area of Octagon?

The area of an octagon is 2s2(1+√2). By using the following steps mentioned below we can find the area of the octagon.

  • Step 1: Calculate the length of the side of the octagon.
  • Step 2: Find the square of the length of the side.
  • Step 3: Find out the product of the square of its length to 2(1+√2). This will give the area of the octagon.
  • Step 4: By substituting the respective values in the area of an octagon formula 2s2(1+√2) we will get the answer.
  • Step 5: Represent the answer in square units.

Area Of An Octagon Formula

You know that polygons are 2D closed figures which are made up of 3 or more line segments. Triangle is the smallest polygon with 3 sides. Similarly, the polygon which has 8 sides is called an octagon. If the 8 sides of the octagon are equal in length and the angles between these sides are equal, the octagon is called a regular octagon. In a regular octagon, the measure of each of the interior angle is 135 degrees. The exterior angles measure 45 degrees each.

In this section, you will learn how to calculate the area of an octagon. Also, how to derive a formula for the same and solve some examples so as to give you a clear understanding of the concept.

Now, the formula for computing the area of a regular octagon can be given as:

Area of an Octagon = 2a2(1+√2)2a2(1+2)

Where ‘a’ is the length of any one side of the octagon.

Click here to learn more about octagon and its types in detail.

Properties of a Regular Octagon:

Before we go ahead and derive the area of an octagon formula, let us go through some of the basic properties of a regular octagon.

  • It has 8 sides and 8 interior angles.
  • All the sides are equal in length and all the angles are equal in measurement.
  • There are a total of 20 diagonals in a regular octagon.
  • Each interior angle measures 135 degrees. Hence, the sum of all interior angles is (135 x 8) = 1080 degrees.
  • Each exterior angle measures 45 degrees. Hence, the sum of all exterior angles is (45 x 8) = 360 degrees.

Properties of All octagons

Basis

Property

Number of diagonals

20

The number of well-defined, recognizable diagonals possible from all vertices (usually ½ n(n–3)].

Number of triangles

6

The number of triangles formed by constructing the diagonals from a given vertex (usually n–2).

Sum of interior angles

1080°

Usually 180(n–2) degrees.

How to Calculate the Area of a Regular Octagon

You can calculate the area of a regular octagon with the standard regular polygon method, but there’s a nifty alternative method based on the fact that a regular octagon is a square with its four corners cut off.

For example, here’s how you’d find the area of EIGHTPLU in the figure below, given that it’s a regular octagon with sides of length 6.

The four corners (like triangle SUE in the figure) that you cut off the square to turn it into an octagon are 45°- 45°- 90° triangles (you can prove that to yourself if you feel like it). So all you have to do to get the area of the octagon is to calculate the area of the square and then subtract the four corner triangles. Piece o’ cake.

But first, here are two great tips for this and other problems.

  • For problems involving regular octagons, 45°- 45°- 90° triangles can come in handy. Add segments to the diagram to get one or more 45°- 45°- 90° triangles and some squares and rectangles to help you solve the problem.
  • Think outside the box. It’s easy to get into the habit of looking only inside a figure because that suffices for the vast majority of problems. But occasionally (like in this problem), you need to break out of that rut and look outside the perimeter of the figure.

Okay, so here’s what you do. You’re given that the octagon’s sides have a length of 6. Consider side EU.

Not just calculate the area of the square and the area of a single corner triangle:

To finish, subtract the total area of the four corner triangles from the area of the square:

Solved Examples on Area of Octagon

Formula for Perimeter of an Octagon:

Perimeter of an octagon is defined as the length of the boundary of the octagon. So perimeter will be the sum of the length of all sides. The formula for perimeter of an octagon is given by:

Perimeter = length of 8 sides

So, the perimeter of an Octagon = 8a

Properties of a Regular Octagon:

  • It has eight sides and eight angles.
  • Lengths of all the sides and the measurement of all the angles are equal.
  • The total number of diagonals in a regular octagon is 20.
  • The sum of all interior angles is equal to 1080 degrees, where each interior angle measures 135 degrees.
  • The sum of all exterior angles is equal to 360 degrees, where each exterior angle measures 45 degrees.

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 Greek means eight, so it is safe to assume an octagon has eight sides, or that an octopus has eight legs.

We will 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 have?” the answer is always the same, but if they ask about the length of those sides, there is not one answer we can give. 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.

Octagon angles: What is an octagon?

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 straight lines. The eight sides 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 the question what is an octagon? is equally well answered by saying “an eight-sided polygon” as well as “a polygon with eight angles”.

While the sides of an 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 impossible to join all eight sides together without following this rule. For any octagons, the sum of all of its internal angles is always 1080°.

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

Area of an octagon: How to find the area of 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:

area of regular polygon = perimeter * apothem / 2,

where the apothem is the distance for the center of the polygon to the mid-point of a side. 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 remember this formula is to understand where it comes from. If you look at the apothem you can see 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 area is:

area of triangle = base * height / 2

as we explained in the triangle area calculator. Note that the base of the triangle is the length of a side of the octagon. Since there are as many of these triangles as the polygon has sides (eight for an octagon), you have to multiply the area of this triangle by the number of sides. You will obtain the total area of the octagon:

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

These tricks work for any polygon, e.g., hexagons and any other polygon you can think of, as long as it is regular. Apart from using triangles, there are 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. “How to find the area of a regular octagon with this information?” you might ask. Well, it is very easy:

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

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 the parallelogram you just made from the four right triangles, and subtract it from the 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, since both trapezoids are equal.

Diagonals of a regular octagon

In total, an octagon has 20 diagonals; the longest ones are on its axes of symmetry and they meet at the central 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):

  • Short diagonals, for example, AG, FH or CE;
  • Medium diagonals, such as AF or BE (also called the height of an octagon);
  • Long diagonals, for example, AE or BF.

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:

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

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

Circumradius and inradius

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.

You can notice 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, which is just half of the octagon’s height:

r = m/2 = a/2 * (1 + √2)

Octagon shape: How to draw an octagon?

It might seem easy to draw an octagon at first, and, 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 octagon, as its is the octagon shape that comes to mind first. So let’s see how to draw an octagon as regularly and as easy as possible.

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

With the circumference drawn, start dividing the corresponding [circle area](calc:752) into halves. First, you will make two half-circles. Then, divide them in half again and get four quarter circles. Halve it once more and you end up with **eight circle-eights**. We can now take the points of division along the circumference and join the adjacent ones with a straight line. The result is a perfectly regular octagon, minus the human error you may make whilst drawing and halving.

Another how to draw an octagon trick would be the one we have already discussed when learning how to find the area of an octagon with squares. Start by drawing 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.

How to use the octagon calculator

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.

  • 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 by 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 2 vertices with 2 vertices in between them
  • Circumcircle radius – Radius of the circumference that contains all 8 vertices of the regular octagon
  • Incircle radius – Radius of the smallest circumference that is tangent to all 8 sides

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.

Octagons in real life: the octagon house

So far we have talked about 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 have. Now, it is time to see how octagons are used in real life. The octagon shape is easier to 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.C. in the USA. It was built for Colonel John Tayloe III. This is the reason, it’s also known as 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 headquarters of the United States Department of Defence and 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 the obvious octagonal shape of the building. On top of such unique characteristics, there are several ghost stories surrounding the house, partly because the owner was an important figure in the 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 particularly popular when it comes to flooring, mostly 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!

Another interesting use of octagons is inside photographic cameras. In particular, the shutter that 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 it to get more professional looking pictures with our [aperture area calculator](calc:1027).

Before you head out to check that calculator we want to give you a quick tip/trick. When you see a picture of a 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 sides of the polygon that makes up the aperture. And how many sides does an octagon have? Exactly! So you only need to look for ‘star shapes’ with 8 pointy arms and you’ll know that this picture was taken using an octagonal aperture. It works every time – it’s physics, and, more specifically, a diffraction pattern.

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