Hm, Octagon Calculator, seems interesting! We believe you’ll discover an outstanding tool for calculating a regular octagon’s diagonals, perimeter, circumradius, inradius, and area. You’ll also discover answers to queries like an octagon, how many sides an octagon has, how to determine the area of a standard octagon, and how to draw an octagon in this octagon area calculator. Next, we’ll go through the definition of an octagon and the octagon angles and how they affect the form. On top of that, we’ll see a few real-world octagon instances. While you are here, check our Pentagon Calculator.

What is an octagon?

An octagon is an eight-sided polygon or 8-gon in geometry. A regular octagon bears the Schläfli symbol and may be made into a quasiregular truncated square with two types of edges. A hexadecagon is a truncated octagon. If one considers the octagon a truncated square, a 3D counterpart of the octagon can be the rhombicuboctahedron with triangular faces on it as replacement edges.

Octagon angles

The presence of eight angles necessitates the presence of eight sides. In truth, the traditional octagon definition calls for eight sides rather than eight angles, but you already know that the two definitions are almost identical.

While the sides of an octagon can be virtually any length, the angles of the octagon are limited. There is a constraint on the aggregate of the angles, not on each one. This constraint is geometric since it would be impossible to connect all eight sides without adhering to this rule. Each inner angle of a normal octagon is 135°. The inside angles add up to 1080°, whereas the external angles add up to 360°.

Sum \; of  \; Interior \; Angles = 135° \times 8 \; sides = 1080°
Sum \; of  \; Exterior \; Angles = 45° \times 8 \; sides = 360°

Octagon definition

The octagon is a two-dimensional geometrical shape with eight sides. There are eight internal angles and eight outside angles in an octagon. The total of an octagon’s inside angles is 1080°, whereas the sum of its outside angles is 360°.

In an octagon, there are 20 diagonals. The sides and angles are used to classify it into several varieties.

A polygon having eight sides and eight inner angles are known as an octagon. The term “octagon” comes from the Greek word “oktágnon,” which means “eight angles.” It is termed an octagon for this reason.

How many sides does an octagon have?

A polygon having eight sides, eight internal angles, and eight vertices are known as an octagon. An octagon is considered a regular octagon when all sides and angles have the same dimension.

Every polygon has a convex or concave shape. Concave octagons have indentations, whereas convex octagons flare outwards (a deep recess). Convex octagons have all of their angles pointing outwards. A convex octagon is similar to a conventional octagon. A concave octagon is one in which at least one of its angles points inwards.

Types of Octagons

An octagon is categorized into the following groups based on its sides and angles:

  • Regular and Irregular Octagon
  • Concave and Convex Octagon

Regular Octagon

A regular octagon is an octagon with eight congruent sides and angles. All of the angles of a regular octagon are equal.

  • All of the sides of a regular octagon have the same length, and all of the angles are the same size.
  • The inner angles total 1080°, whereas the external angles total 360°.
  • A regular octagon’s internal angle at each vertex is 135°.

Irregular Octagon

An irregular octagon is one in which the sides and angles are not congruent. An irregular Octagon, in other terms, has eight uneven sides and eight unequal angles.

  • It’s an octagon with unbalanced angles and sides.
  • The internal angles are all measured differently, but the sum is always 1080o.

Convex Octagon

The octagon is convex because all of its angles point outwards and none point inwards. A convex octagon’s angles are all less than 180 degrees.

  • The octagons of convex octagons flare outwards.
  • None of their inner angles exceed 180 degrees.

Concave Octagon

A concave octagon is one in which one of the angles points inward.

  • We find the indentations on concave octagons (a deep recess).
  • At least one of the inner angles is bigger than 180°, indicating that it is a reflex angle.

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

The simplest method for calculating the area of a regular octagon by hand is to use the usual formula for calculating the area of a regular polygon. The formula is as follows, where the apothem is the distance between the polygon’s center and the mid-point of one of its sides:

Area \; of \; Regular \; polygon = perimeter \times \frac {apothem} {2}

Understanding where this formula originates from is a key to memorizing it. When you look at the apothem, you’ll notice that it’s the height of a triangle formed by drawing a line from the polygon/vertices octagons to the center. The resultant triangle is an isosceles triangle, with the following area:

Area = base \times \frac {height} {2}

The triangle’s base is equal to the length of one of the octagon’s sides. Because there are as many triangles as the polygon’s sides (eight for an octagon), the area of this triangle must be multiplied by the number of sides. You’ll get the octagon’s entire area as follows:

Octagon \; Area = 8 \times base \times \frac {height} {2} = perimeter \times \frac {apothem} {2}

There is no exact formula for calculating the area of an irregular octagon. Instead, the octagon is divided into smaller forms, such as triangles. After computing the area of each triangle, we sum their areas to get the octagon’s area.

Diagonals of a regular octagon

An octagon contains a total of 20 diagonals, the longest of which are on its symmetry axis and intersect at the center point O, which is also the origin of symmetry. There are three different kinds of diagonals.

There are three types of diagonals:

  • short,
  • medium, and
  • long.

Using fundamental geometric concepts, you can easily get the formula for each. For example, we may calculate the length of the diagonals (short – s, medium – m and long – l) using the following formulas:

s = a \cdot \sqrt {2 + \sqrt {2}}
m = a \cdot (1+\sqrt {2})
l = a \cdot \sqrt {4+2 \cdot \sqrt {2}}

Circumradius and inradius

The circumradius is just half the length of the largest diagonal, as seen below:

R = \frac {l} {2} = \frac {a} {2} \cdot \sqrt {4+2 \cdot \sqrt {2}}

The inradius is the same as the apothem, which is half the height of the octagon:

r = \frac {m} {2} = \frac {a} {2} \cdot (1+\sqrt {2})

Octagon shape: How to draw an octagon?

At first, glance, drawing an octagon may appear simple, and it is. Simply draw any form with eight straight sides to complete the task. However, most people prefer to draw a conventional octagon since it’s the first form that comes to mind when they think of an octagon. So, let’s examine how to draw an octagon in the most orderly and simple manner possible.

Steps in drawing an octagon

The most exact method would be to start creating the regular octagon form one side at a time, linking them with the correct 135° regular octagon angles, using suitable drawing tools. However, hardly everyone has such instruments on hand, making this strategy impractical. So instead, to get the octagon form we want, start with the circumference. Which you may sketch by hand (which we don’t suggest) or using a glass, cup, or even a coin.

Begin splitting the relevant circular area into half using the circumference drawn. You’ll start by making two half-circles. After that, cut them in half again to get four quarter circles. Finally, cut it in half again, and you’ll get eight circle-eights. We may now take the division points throughout the perimeter and connect them straight. The result is an octagon that is exactly regular, minus whatever human errors you may have made during sketching and halving.

How to use the octagon calculator

The octagon calculator is simple to use, but just in case anyone has any questions or wants to be sure, let’s go over the features and applications of this octagon area calculator. This calculator’s finest feature is that it just takes one input to calculate the remaining numbers and obtain desired results. By a large margin, this invention makes the octagon calculator the fastest way to calculate any attributes of an octagon.