## Polygons – Octagons

Octagonal properties and internal angles of octagons are discussed in detail.

Polygons:Properties of Octagons |

Sum of the Interior Angles of an Octagon: |

This image shows the process for aHEXAGON: | Using the same methods as for hexagonsto the right(I’ll let you do the pictures).To find the sum of the interior angles of an octagon, divide it up into triangles. There are six triangles.Because the sum of the angles of each triangle is 180 degrees.We getSo, the sum of the interior angles of an octagon is 1080 degrees. |

Regular Octagons: | |

The properties of regular octagons: | |

All sides are the same length (congruent) and all interior angles are the same size (congruent).To find the measure of the angles, we know that the sum of all the angles is 1080 degrees (from above).And there are eight angles.So, the measure of the interior angle of a regular octagon is 135 degrees. | |

The measure of the central angles of a regular octagon: | |

To find the measure of the central angle of a regular octagon, make a circle in the middle.A circle is 360 degrees around.Divide that by eight angles.So, the measure of the central angle of a regular octagon is 45 degrees. |

## Interior Angles of Polygons

Here’s another illustration:

## Triangles

The sum of the interior angles of a triangle equals 180°. Let’s try a triangle for a change: 180° is equal to 90° plus 60° plus 30°. It is effective in this triangle. Tilt a line by 10 degrees now: 180° is equal to 80° plus 70° plus 30°. It is still operational! One angle increased by 10°, while the other decreased by 10°.

## Quadrilaterals (Squares, etc)

(A quadrilateral is a shape with four straight sides.) Let’s attempt a square this time: 360° is equal to 90° plus 90° plus 90° plus 90°. The sum of a square is 360 degrees. Tilt a line by 10 degrees now: 80° plus 100° plus 90° plus 90° equals 360°. It still adds up to a whole 360 degrees. Quadrilaterals have interior angles that sum up to 360 degrees.

### Because there are 2 triangles in a square.

The internal angles of a triangle sum up to 180°. and the interior angles of a square add up to 360°. because a square may be formed by joining two triangles together!

## Pentagon

Because a pentagon has five sides and can be constructed from three triangles, the internal angles of a pentagon sum up to three 180° angles, or 540°. If the pentagon is regular (all angles are the same), then each angle is equal to 540°/ 5 = 108° (Example: verify that each triangle here adds up to 180°, and check that the pentagon’s inner angles sum up to 540°). All of the inside angles of a Pentagon come together to equal 540°.

## The General Rule

If we add a side (a triangle to a quadrilateral, a quadrilateral to a pentagon, and so on), we increase the total by another 180 degrees: As a result, the general rule is as follows: The sum of interior angles equals (n2) 180 degrees. For each angle (of a Regular Polygon), the formula is: (n 2/180 °/n Perhaps the following illustration will be of assistance:

### Example: What about a Regular Decagon (10 sides)?

In the sum of interior angles, (n 2) 180 degrees equals (102) 180 degrees equals 8 180 degrees equals 1440 degrees. And for a Regular Decagon, the following is true: Each inner angle is equal to 1440/10 =144°. Please keep in mind that inside angles are frequently referred to as “Internal Angles.”

## Octagon – Wikipedia

Regular octagon | |
---|---|

A regular octagon | |

Type | Regular polygon |

Edgesandvertices | 8 |

Schläfli symbol | , t |

Coxeter–Dynkin diagrams | |

Symmetry group | Dihedral(D 8), order 2×8 |

Internal angle(degrees) | 135° |

Properties | Convex,cyclic,equilateral,isogonal,isotoxal |

In geometry, an anoctagon (from the Greek voktágnon, “eight angles”) is a polygon with eight sides, sometimes known as an 8-gon. This shape is represented by the Schlifli symbol and may alternatively be formed as a quasiregulartruncatedsquare, denoted by the letter t, which alternates between two types of edges. Ahexadecagon is an octagon that has been truncated. If one considers the octagon to be a truncated square, a 3D analog of the octagon may be represented by therhombicuboctahedron, which has triangle faces on it that act as replacements for the edges of the octagon.

## Properties of the general octagon

Equal in length and at right angles to one another, the diagonals of the green quadrilateral form a right triangle. The total of all of the internal angles of every octagon equals 1080° in all directions. As with other polygons, the exterior angles add up to a total of 360 degrees. If squares are constructed entirely internally or entirely externally on the sides of an octagon, then the midpoints of the segments connecting the centers of opposite squares form a quadrilateral that is bothequidiagonalandorthodiagonal(that is, whose diagonals are equal in length and at right angles to each other).: Prop.

10

## Regular octagon

The aregularoctagon is a closed figure with sides that are all the same length and internal angles that are all the same size as one another. In all, it possesses eight lines of reflectional symmetry and rotational symmetry of order 8 in its structure. TheSchläfli sign represents a regular octagon in its simplest form. One hundred and thirty-five degrees are included within each vertex of a regular octagon (radians). The middle angle is 45 degrees (radians).

### Area

The area of a regular octagon with equal side lengths is given by the formula The area is measured in terms of the circumradiusR. When seen via the lens of theapothemr(see also engraved illustration), the region is These final two coefficients are used to frame the value of pi, which represents the area of the unit circle. It is also possible to represent the area in the following way:whereS is the span of the octagon, or the second-shortest diagonal; anda is the length of one of the sides, or bases To demonstrate this, take an octagon and draw a square around the outside (ensuring that four of the eight sides overlap with the four sides of the square), and then take the corner triangles (which are 45–45–90 triangles) and place them with right angles pointed inward, resulting in the formation of a square.

The length of each of the square’s edges equals the length of the base.

The span, on the other hand, is equal to the silver ratio multiplied by the length of the side, a.

Another straightforward formula for the region is It is more typically the case that the spanSis known, and it is necessary to calculate the length of the sidesa, as when cutting a square piece of material into a regular octagon.

### Circumradius and inradius

The circumradiusof a regular octagon is defined by the lengths of its sides and the radius of its inradius (that is one-half thesilver ratiotimes the side,a, or one-half the span,S)

### Diagonals

There are three different types of diagonals in the regular octagon, depending on the length of its sides:

- A short diagonal line
- The medium diagonal (also known as the span or the height) is twice the length of the inradius
- And Long diagonal, which is twice as long as the circumradius
- Long diagonal that is twice as long as the circumradius

Each of them has a formula that is derived from the fundamental principles of geometry. The following are the formulae for determining their length:

### Construction and elementary properties

By folding a piece of paper in half, you can create a regular octagon. The following is an example of how to create a regular octagon at a given circumcircle:

- A circle with a diameter AOE, where O is the center and A, E are points on the circumcircle, are drawn. Draw a second diameter GOC that is perpendicular to the AOE. (It should be noted that the letters A, C, E, and G represent the vertices of a square.) Draw the bisectors of the right angles GOA and EOG, resulting in two additional diameters HOD and FOB
- The vertices of the octagon are A, B, C, D, E, F, G, and H.

At a certain circumcircle, an octagon appears. It is possible to make a regular octagon with a straightedge and a compass, because 8 = 2 3 is a power of two: Construction of an octagon with Meccano pieces. Meccanobars may be used to build a regular octagon, which can be found here. It is necessary to use twelve bars of size 4, three bars of size 5, and two bars of size 6 in total.

Each side of a regular octagon subtends half of a right angle in the center of the circle that connects its vertices, resulting in a total of eight right angles. In this case, the area may be calculated as the sum of eight isosceles triangles, yielding the following result: for an octagon of sidea.

### Standard coordinates

The coordinates for the vertices of a regular octagon centered at the origin and having a side length of 2 are given by the following formula:

### Dissection

8-cubeprojection | 24 rhomb dissection | |
---|---|---|

Regular | Isotoxal | |

In his paper, Coxeter asserts that everyzonogon (a 2m-gon whose opposing sides are parallel and of equal length) may be divided into a pair of parallelograms of lengthm (m -1). This is especially true for regular polygons with an equal number of sides, in which case the parallelograms are all rhombi. One example of this is shown below. For the normal octagon,m = 4, and it may be split into 6 rhombs, as illustrated in the diagram below. Six of the twenty-four faces in a Petrie polygonprojection plane of the tesseract may be represented by this decomposition.

The Ammann–Beenker tilings make use of squares and rhombuses like this one.

Regular octagon dissectedTesseract | 4 rhombs and 2 square |

## Skew octagon

In this picture, the edges of an antiprism with symmetry D 4d, (2*4) and order 16 are shown as the edges of a regular skew octagon. The askew octagon is a polygon with 8 vertices and edges that does not exist on the same plane as the other polygons. However, the inside of an octagon of this type is not typically defined. Each of the vertices of an askew zig-zag octagon alternates between two parallel planes. The octagon with a regular skew is vertex-transitive and has equal edge lengths. A zig-zag skew octagon will be formed in three dimensions, and its vertices and side edges will be seen in the vertices and side edges of a square antiprism with the same D 4d,symmetry, order 16 as a square antiprism.

### Petrie polygons

This higher-dimensional regular and uniform polytopes are represented by the regular skew octagon, which can be seen in these skeworthogonal projectionsof in the A 7, B 4, and D 5Coxeter planes. The regular skew octagon is also known as the Petrie polygon.

A 7 | D 5 | B 4 | |
---|---|---|---|

7-simplex | 5-demicube | 16-cell | Tesseract |

## Symmetry of octagon

SymmetryThe 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. |

Theregular octagons have Dih 8symmetry and are ordered 16 times. A total of three dihedral subgroups (Dih 4, Dih 2, and Dih 1) exist, as well as four cyclic subgroups (Z 8, Z 4, Z 2, and Z 1), the last of which implies no symmetry.

Example octagons by symmetryr16 | ||
---|---|---|

d8 | g8 | p8 |

d4 | g4 | p4 |

d2 | g2 | p2 |

a1 |

On a normal octagon, there are 11 different symmetries to consider. Asr16 is the designation given by John Conway to complete symmetry. The dihedral symmetries are classified into two groups based on whether they travel through vertices (d for diagonal) or edges (e for edge-passing) (pfor perpendiculars) The core gyration orders of the cyclic symmetries in the middle column are denoted by the letters asg. The regular form has complete symmetry, which is labeledr16, and it has no symmetry, which is labeleda1.

These two shapes are duotones of one another and have half the symmetry order of a standard octagon, respectively.

Each subgroup symmetry provides for one or more degrees of freedom in the creation of irregular shapes. Only theg8subgroup has no degrees of freedom, yet it may be viewed as having directed edges because of this.

## Uses of octagons

Dome of the Rock’s floor layout is octagonal in shape. In architecture, the octagonal form is frequently employed as a design feature. TheDome of the Rockhas an octagonal shape that distinguishes it from other structures. Another octagonal building is theTower of the Windsin Athens, which was built in the early 1900s. St. George’s Cathedral in Addis Abeba, the Basilica of San Vitale in Ravenna, Italy, the Castle of Monte in Apulia, Italy, the Florence Baptistery, the Zum Friedefürsten Church in Germany, and a number of octagonal churches in Norway are examples of the use of the octagonal plan in architectural design and architecture.

Minor architectural features like as the octagonal apex of Nidaros Cathedral’s octagonalapse are also included into octagon-based church architecture.

### Other uses

- The octagonal “elephant’s foot” pattern is used prominently in the famousBukhara rug design. The streetblock pattern of Barcelona’s Eixampledistrict is built on non-regular octagons
- This is known as the “Eixampledistrict layout.” In the centre of the sign is a hand, which represents a stop. The trigrams of theTaoistbagua are frequently grouped in an octagonal pattern. Shimer College’s classes are generally held around octagonal tables, as is customary.

## Derived figures

Theoctagon, in the form of an atruncatedsquare, is the first in a sequence of truncatedhypercubes, which includes:

Truncated hypercubesImage | . | |||||||
---|---|---|---|---|---|---|---|---|

Name | Octagon | Truncated cube | Truncated tesseract | Truncated 5-cube | Truncated 6-cube | Truncated 7-cube | Truncated 8-cube | |

Coxeter diagram | ||||||||

Vertex figure | ()v() | ()v | ()v | ()v | ()v | ()v | ()v |

As an extended square, it is also the first in a series of expanded hypercubes, which are as follows:

Expanded hypercubes. | |||||||

Octagon | Rhombicuboctahedron | Runcinated tesseract | Stericated 5-cube | Pentellated 6-cube | Hexicated 7-cube | Heptellated 8-cube | |

## See also

- Bump pool
- Octagon house
- Octagonal number
- Octagram
- Octahedron, a 3D structure with eight faces
- Bumper pool
- Oktogon is a prominent crossroads in the Hungarian capital of Budapest. Rub el Hizb (also known as the Al Quds Star and the Octa Star)
- Smoothed octagon
- Rub el Hizb (also known as the Al Quds Star and the Octa Star)

## References

- AbDao Thanh Oai (2015), “Equilateral triangles and Kiepert perspectors in complex numbers,” Forum Geometricorum 15, 105-114
- AbDao Thanh Oai, “Equilateral triangles and Kiepert perspectors in complex numbers,” Forum Geometricorum 15, 105-114
- AbDao Thanh Oai, “Equilateral triangles and Kiepert perspectors in

## External links

Look upoctagonin Wiktionary, the free dictionary. |

- Octagon Calculator
- Definition and attributes of the octagon with interactive animation
- Octagon Calculator

## How to Figure Degrees in an Octagon

An octagon is a form with eight sides, such as the shape of a stop sign. Octagons can be either regular or irregular in shape. Normally, the sides of an octagon are congruent, meaning they are all the same length. An irregular octagon is characterized by the presence of sides of varying length. As soon as you’ve figured out how many degrees each angle has in total, knowing whether the octagon is regular or irregular can assist you in determining the measure of each individual angle in the octagon.

To figure out the unknown eighth angle in an irregular octagon, you must first work out the other seven angles that make up the irregular octagon.

## Regular Octagons

- Add two to the number of sides in an octagon to get the total number of sides. Because an octagon has eight sides, you can subtract two from eight to get the number six. To get the total number of degrees in an octagon, multiply six by 180 and divide the result by six. The result is 1,080 degrees. The measure of each inner angle in an octagon that is regular may be calculated by dividing 1,080 by eight. Each angle in a standard octagon measures 135 degrees
- However, in a distorted octagon, each angle measures 135 degrees.

## Irregular Octagons

- Add two to the number of sides in an octagon to get the total number of sides. Because an octagon has eight sides, you can subtract two from eight to get the number six. To get the total number of degrees in an octagon, multiply six by 180 and divide the result by six. The result is 1,080 degrees. To calculate the sum of the seven known angles, add the angle measurements of the seven known angles together. For example, if your seven known angles have measurements of 100, 110, 120, 140, 150, 160, and 170, the total of the angles is 950, as shown in the diagram. If you have an irregular polygon, you may get the measure of the unknown angle by subtracting the measure of the seven known angles from 1,080 to find the measure of the known angles. As a last step, subtract 950 from 1,080 to determine the unknown angle, which is 130 degrees.

- If you do not have access to the angles that have been provided to you, you can calculate the angle measurements using a protractor. To use a protractor, start with the origin above the angle vertex and move the protractor until it is aligned with one of the angles sides. Next, figure out what the degree measure is based on the point at which the second side of the angle crosses the angle measurement on the protractor.

## Octagon

- N internal angles
- The sum of interior angles equals (n – 2) 180°
- N interior angles Each internal angle of a regular polygon is equal to
- The sum of all exterior angles is equal to 360°.

Throughout this article, we will study everything there is to know about the eight-sided polygon known as the “octagon,” including its proper definition as well as its shape, number of sides, attributes, and formulas for perimeter and area.

### Definition of Octagon

(Illustration will be updated shortly.) An octagon is a polygon with eight sides and eight angles that is made up of four triangles. Eight angles are represented by the word “octagon,” which is composed of two words, namely “octa” and “Gonia,” which are both derived from Latin. Because the octagon has eight sides, we may say that = (8 – 2) 180° = 6 180° = 1080° = 8 – 2 180°

### Shapes of Octagon

Octagon forms are divided into the following categories based on their sides, angles, and vertices:

### Regular Octagon

(Illustration will be updated shortly.) To qualify as a normal octagon, the octagon must contain the following characteristics:

- Eight congruent sides (sides that have of the same length)
- Eight inner angles that are congruent (each measuring 135°)
- Eight 45-degree external angles that are congruent

It is important to note that regular octagons do not have parallel sides. The octagon is distorted. Unusual octagons are the octagons with a variety of side lengths and angle measurements. (Illustration will be updated shortly.) Irregular octagons can be either convex or concave, depending on their shape:

- A convex octagon is an octagon that does not have any interior angles greater than 180 degrees. A concave octagon is an octagon with an internal angle greater than 180° on one side.

(Illustration will be updated shortly.)

### Properties of Octagon

- It has eight sides, eight vertices, and eight interior angles
- It also has eight internal angles. It has a total of 20 diagonals. Ten hundred eighty degrees is the total of all internal angles. The total of all of the outer angles is 360 degrees. A regular octagon has eight sides that are of the same length
- An octagon with uniform angles has 135 degrees of internal angle on each side. Uneven octagons have differing side lengths and angle measurements than ordinary octagons. While all of the diagonals of the convex octagon are contained within the octagon, some of the diagonals of the concave octagon may be found outside the octagon.

### The Perimeter of an Octagon

The perimeter of an octagon is equal to the sum of the lengths of the eight sides of the shape it represents. Because the lengths of all eight sides of a regular octagon are equal, this is true for a normal octagon. Thus, the perimeter of a regular octagon is equal to 8 (side length) units of space.

### Area of an Octagon

The region covered by the sides of the octagon is referred to as the area of the octagon. The area of a regular octagon may be computed using the formula: (image will be updated soon)

### If the Measure of Side Length and Apothem is Given, Then:

In mathematics, an apothecary is a line that runs from the center of a regular polygon at right angles to any of its sides. In either case, the area of an octagon is equal to (side length) x (apothem) units2 OR the area of an octagon is equal to (perimeter of an octagon) x (apothem) units2.

### If the Only Measure of Side Length is Given, Then:

The area of an octagon is equal to 2(1 +2–2) (side length)2units2.

### Length of the Longest Diagonal of an Octagon

If we link the opposing vertices of a regular octagon, the diagonals created have the length L = (side length) units, and the diagonals formed have the length L = (side length) units. The octagon’s axis of symmetry is defined by the longest diagonals.

### Solved Problems:

1. What is the perimeter and area of a normal octagon with a side length of 7 cm? Solution: Given that the side of the octagon is 7 cm, The perimeter of a regular octagon is equal to 8 (side length) units, which is equal to 8 7 cm, which is equal to 56 cm. And, The area of an octagon is equal to 2(1 +2–2) (side length)2 units. 2= 4.828 (7)2= 236.572 cm2 2= 4.828 (7)2= 236.572 cm2 2= 4.828 (7)2= 236.572 cm2 2= 4.828 (7)2= 236.572 cm2 2= 4.828 (7)2= 236.572 cm2 2= 4.828 (7)2= 236.572 cm2 2= 4.828 (7)2= 236.572 cm2 2= 4.828 (7)2= 236.572 cm2 2 As a result, the circumference and area of a regular octagon with a side length of 7 cm are 56 cm and 236.572 cm2 respectively, when the side length is 7 cm.

- Solution: Given that the side of the octagon is 8 cm, The length of the longest diagonal of a regular octagon is equal to the number of units in the side length.
- When it comes to geometry, octagons are a type of polygon with eight sides and eight angles that may be explored.
- These notes, which are prepared by Vedantu’s expert research team, have years of experience and are well versed in the subject matter.
- Vedantu provides students with the most up-to-date material that is necessary in order to get a good score in the examination.
- It is prepared in an exceedingly simple style so that students who may find it difficult to grasp the complicated topics included in the NCERT book and read through the notes supplied by Vedantu would be able to understand them.
- When drawn in a two-dimensional plane, an octagon has eight vertices and eight edges; hence, an octagon is regarded to be an eight-sided polygon, also known as an 8-gon.
- The measurement of each inner angle of an octagon is 135°, which gives us the total measurement of all the outer angles of an octagon, which are each 45°.
- It is a two-dimensional figure with a closed top and bottom.

The term “regular octagon” refers to an octagon that has all of its sides, internal angles, and exterior angles equal to one another; otherwise, the term “irregular octagon” is used. The other forms of octagons are convex and concave octagons.

### Examples of an octagon in our day to day lives include –

A wall clock with eight edges, stop sign boards at traffic lights, and other like things. An octagon is also represented by the shape of an umbrella with eight ribs.

### Properties of an octagon –

The qualities mostly correspond to those of a regular octagon, and they are as follows: Regular octagons have eight sides and eight angles, much like the rest of the world. In an octagon, the total of all of its outside angles is 360 degrees, with each angle measuring 45 degrees. An octagon with uniform angles has a total interior angle of 1080°, which is the sum of all the interior angles. The very forest measures 135 degrees. The diagonals of a normal octagon are 20 in number. A regular octagon has exactly the same number of sides and angles as it has corners.

### Types of an octagon-

There are four different forms of octagons: the regular octagon, the irregular octagon, the concave and convex octagon, and the hexagon. An octagon with regular sides has previously been examined in detail in this section. An irregular octagon is an octagon with uneven sides and unequal angles, which is defined as follows: a convex octagon is an octagon that has all of its angles facing outwards and none of its angles pointing inwards is referred to as such. A convex octagon has angles that are fewer than 180 degrees.

### Certain concepts that are related to the study of octagons are as follows-

- The characteristics of the generic octagon
- Regular octagon
- Regular octagon
- Regular octagon The area of an octagon
- The circumradius and inradius of an octagon
- The radius of an octagon Octagonal diagonals
- Octagonal diagonals Building an octagon and learning about its fundamental properties The octagon’s standard coordinates are as follows: An octagon is dissected in detail. octagon with a skewed angle
- Polygons of Petrie’s kind
- The symmetry of the octagon
- The usefulness of octagons
- And other applications Figures derived from other sources

## Angles to Cut Octagonal Pieces

I’m planning on building an octagonal lighthouse, and I’m having trouble determining the angle at which to cut the parts. Scott Noyes (interviewer): The angle at which each component is to be glued is 67.5 degrees. The interior angles of a regular polygon may be calculated using the formula A (angle) = 180 – (360/n), where n is the number of sides of the polygon. As a result, the internal angles of an octagon each measure 135 degrees. (360 multiplied by eight equals 45. (135 is obtained by subtracting 45 from 180).

- For this, you either split the angle in half to obtain 67.5 degrees, or you can apply the formula 90 – (180/n), which yields the same result as the previous method.
- To find the angle, divide the number of pieces wanted (8) by the number of degrees in a circle (360), and then divide the result by two since half of the angle will be applied to each side of your stock stock.
- Normally, your table saw is placed at 90 degrees, or perpendicular to the top of the table.
- Set the blade tilt at 67.5 degrees.

## Unit 15 Section 2 : Angle Properties of Polygons

In this section we calculate the size of the interior and exterior angles for different regular polygons.In a regular polygon the sides are all the same length and the interior angles are all the same size.The following diagram shows a regular hexagon:Note that, for any point in a polygon, the interior angle and exterior angle are on a straight line and therefore add up to 180°.This means that we can work out the interior angle from the exterior angle and vice versa: |

Interior Angle = 180° – Exterior Angle | Exterior Angle = 180° – Interior Angle |

## What is the sum of the angles of an octagon?

There is a straightforward method for calculating the sum of all the internal angles of an octagon, or any polygon for that matter. Make a rough sketch of an octagon (it doesn’t have to be flawless) in order to understand how it works. Now, choose one of the octagon’s corners and draw a line connecting each of the other corners of the octagon to that corner. In this case, you won’t need to draw lines from the corners closest to the one you selected since those lines are already present – they’re two sides of your octagon!

- You should be able to see six non-overlapping triangles that have been glued together to form an octagon.
- The entire internal angle sum of the octagon may be calculated by adding all of the angles in those six triangles together.
- If you want to accomplish this with any convex polygon, you may do it with any convex polygon, where convex means that all of the internal angles are smaller than 180 degrees.
- This is so common that it has been formalized as a theorem: If a convex polygon hasnsides, then the internal angle sum (S) of the polygon may be calculated using the following formula: S is equal to (n– 2) 180°.

In the absence of any additional information, you may use this equation to determine the interior angle total of polygons with 37 side (6300 degrees), 73 side (12,780 degrees), or even 100 side (17,640 degrees).

## Polygons – Angles, lines and polygons – Edexcel – GCSE Maths Revision – Edexcel

In two-dimensional geometry, a polygon is a form with at least three sides.

## Types of polygon

Polygons can be either regular or irregular in shape. A regular polygon is defined as one in which all of the angles are equal and all of the sides are equal in length.

## Interior angles of polygons

The total of internal angles in a polygon may be calculated by dividing the polygon into triangles. An internal angle in a triangle is equal to the sum of its interior angles (180°). Using a polygon as an example, multiply the number of triangles in the polygon by 180° to determine the sum of internal angles.

### Example

In a pentagon, find the sum of the internal angles on each side. A pentagon is made up of three triangles. The total of the internal angles is expressed as follows: The number of triangles in each polygon is two fewer than the number of sides of the polygon in which it is found. To find the sum of interior angles, use the following formula: ((n – 2) times 180circ)(where (n)is the number of sides)(where (n)is the number of sides) Question In an octagon, find the sum of the internal angles on each side.

### Calculating the interior angles of regular polygons

In a regular polygon, all of the inside angles are the same size. The following is the formula for determining the size of an interior angle: Question Calculate the size of the inner angle of a regularhexagon by using the formula below. The total of internal angles is ((6 – 2) times 180 = 720circ) times the radius of the circle. An example of an internal angle is (720 div 6 = 120circ).

## Exterior angles of polygons

When a polygon’s side is stretched, the angle generated outside of the polygon is referred to as the external angle. The sum of the outer angles of a polygon equals 360° in all directions.

### Calculating the exterior angles of regular polygons

The following is the formula for determining the size of an external angle: Keep in mind that the sum of the inner and outside angles equals 180°. Question The size of the exterior and interior angles of a regular pentagon must be calculated. Method 1: The total of all external angles equals 360 degrees. In this case, the outer angle is (360 div 5 = 72circ). The sum of the inner and outside angles equals 180°. 180 – 72 = 108circ is the inner angle of the triangle. Method 2The total of inner angles is ((5 – 2) times 180 = 540circ) ((5 – 2) times 180 = 540circ).

The sum of the inner and outside angles equals 180°.

- An internal angle in a triangle is equal to the sum of its interior angles (180°). A polygon’s sum of its internal angles may be calculated by multiplying the number of triangles in the polygon by 180°. All interior angles in a regular polygon are equal, and the formula for computing the sum of internal angles is((n – 2) times 180circ)where (n) is the number of sides
- All interior angles in a regular polygon are equal. A polygon’s interior angle is calculated using the following formula: interior angle of a polygon = sum of interior angles divided by the number of sides. In a polygon, the sum of its outside angles equals 360°. It is possible to calculate how large an exterior angle is by using the following formula: exterior angle of a polygon = 360 times the number of sides

## Polygon Interior Angles – Math Open Reference

Consider the following: By dragging any orange dot on the polygon below, you may make adjustments. Repeat the process by clicking on “make regular.” Take note of the behavior of the internal angles and the total of their angles. These are the angles at each vertex that are located on the inside of the polygon that are referred to as “internal angles” of a polygon. Each vertex has one of them. As a result, given a polygon of N sides, there are Nvertices and N interior angles to consider. By definition, all of the internal angles of an irregular polygon are the same size.

It’s important to note that for any given number of sides, all of the inside angles are the exact same length.

Each angle of an anirregular polygon may be different from the others. Create an irregular shape by clicking on “make irregular” and experimenting with the number of sides and dragging a vertex.

## Sum of Interior Angles

Inside of any polygon, the interior angles always add up to a constant value, which is dependent solely on the total number of sides of the polygon. It makes no difference if the pentagonalways are regular or irregular, convex or concave, or what size and form they are, the inner angles of apentagonalways sum up to 540°. An interior angle sum of a polygon is determined by the following formula:where is the number of sides in the polygon. As an illustration, consider the following:

Asquare | Has 4 sides, so interior angles add up to 360° |

Apentagon | Has 5 sides, so interior angles add up to 540° |

Ahexagon | Has 6 sides, so interior angles add up to 720° |

. etc |

## In Regular Polygons

Because all of the interior angles have the same values, the total given above is distributed evenly across all of the interior angles of a regular polygon. So, for example, the internal angles of a pentagon always total up to 540°, so each of the five sides of a regular pentagon is one fifth of that, or 108°, in a regular pentagon. Another way of saying it is that each interior angle of a regular polygon is given by:wheren is the number of sides in the polygon

## Adjacent angles

Adjacent interior angles, or simply “adjacent angles,” are two interior angles that share a common side and are referred to as such.

## Other polygon topics

- In geometry, a polygon is defined as a rectangle with four sides that are convex and one that is concave. A quadrilateral is defined as a rectangle with four sides that are concave and one that is convex and one that is concave and one that is convex and one that is concave. Triangles in a polygonal shape
- Apothem of a regular polygon
- The center of a regular polygon The circumference of a regular polygon
- The circumcircle of a regular polygon
- The incenter of a regular polygon
- The circumcircle of a polygon
- The circumcircle of a polygon An inscribed parallelogram within a quadrilateral

### Types of polygon

- Trapezoid, Trapezoid median, Kite, and Inscribed (cyclic) quadrilateral are all examples of polygons. The following are some examples of polygons: square, squares, and rectangles.

- Embedded quadrilateral interior angles
- Embedded quadrilateral area
- Embedded quadrilateral diagonals
- Inscribed quadrilateral interior angles

### Area of various polygon types

- A regular polygon area
- An irregular polygon area
- A rhombus area
- A kite area
- A rectangle area
- An area of a square
- A trapezoid area
- A parallelogram area

### Perimeter of various polygon types

- A polygon’s perimeter (both regular and irregular)
- A polygon’s area
- Triangles have three sides, rectangles have four sides, and squares have five sides. a parallelogram’s circumference is measured in meters
- The circumference of a rhombus
- A trapezoid’s perimeter
- The perimeter of a kite’s perimeter

### Angles associated with polygons

- Angles on the outside of a polygon
- Angles on the inside of a polygon The relationship between inner and exterior angles
- The center angle of a polygon

### Named polygons

- Tetragon has four sides
- Pentagon has five sides
- Hexagon has six sides
- Heptagon has seven sides
- Octagon has eight sides
- Nonagon Enneagon has nine sides
- Decagon has ten sides
- Undecagon has eleven sides
- Dodecagon has twelve sides.

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