# How To Find The Measure Of Each Interior Angle

## Interior Angle Theorem: Definition & Formula – Video & Lesson Transcript

This indicates that all of the sides of the polygon are congruent and all of the internal angles of the polygon are congruent if the polygon is referred to be “regular.” Consequently, you can determine the size of each angle. Consider the octagon that was utilized in the preceding illustration. If the octagon is regular, you can calculate the measure of each angle by dividing the total of the angles by the number of angles in the octagon. The fact that there are eight angles, each of which has the same size, means that each is one-eighth of the total.

Always keep in mind that the total is still 1080 degrees.

Each internal angle of a regular octagon has a measure of 135 degrees, which is the same as its perimeter.

## Finding the Number of Sides

If you know the sum of the interior angles of a polygon, you can use the formula for interior angles to figure out how many sides it has as well. Consider the following scenario: you have a polygon whose internal angles total 540 degrees. When you enter this into the formula, it will be the value ofS, andnwill be the unknown value this time. Due to the fact that the value ofnis five, the polygon is a pentagon.

## Interior Angle Formula (Definition, Examples, & Video) // Tutors.com

VideoDefinition Finding Unknown Angles Using the Sum of Interior Angles Regular Polygons are polygons that are arranged in a regular pattern. If you look at the other geometry classes on this useful website, you will see that we have made a point of including internal angles, rather than just angles, when addressing polygons in our discussions. Every polygon includes both inner and exterior angles, but it is the internal angles that contain the most of the exciting activity and are thus the more engaging.

## What you’ll learn:

After completing this lesson and watching the video, you will be able to do the following:

• Identify the internal angles of polygons by their shape
• Remember and put into practice the formula for calculating the sum of the internal angles of a polygon. Recall a technique for determining an unknown inner angle of a polygon
• And Interior angles of polygons should be calculated. Find out how many sides a polygon has by counting its sides.

## Interior Angle Formula

In a space, polygons close in on each other, starting with the simplest polygon, a triangle, and on to the infinitely complicated polygon withnsides. Every point where two sides meet forms a vertex, and each vertex has an inner and an exterior angle, as shown in the diagram. Interior angles of polygons are those that are contained within the polygon. Despite the fact that Euclid provided an external angles theory that was particular to triangles, there is no Interior Angle Theorem. Instead, you may make use of a mathematical formula that depicts a fascinating pattern using polygons and their internal angles mathematically.

## Sum of Interior Angles Formula

This formula allows you to partition any polygon into the smallest number of triangles possible using only mathematics. Because every triangle contains interior angles that measure 180°, multiplying the number of dividing triangles by 180° gives you the total of the internal angles for that triangle. S=sumofinteriorangles n=numberofsidesofthepolygon Apply the following formula to a triangle: S=(n-2)180° S=(3-2)180° S=1180° S=180° S=(3-2)180° S=1180° S=180° It appears to have succeeded, but what about a more sophisticated form, such as a dodecagon?

Take any dodecagon and choose one of its vertices. With a straightedge, connect every other vertex to that one, splitting the space into ten triangles in the process. Ten triangles, each 180° in angle, add up to a total of 1,800°!

## Finding an Unknown Interior Angle

It is possible to use the same method, S=(n-2)180°, to locate a missing interior angle of a polygon. Here’s an unusual pentagon, with no two sides being the same: According to the formula, a pentagon, no matter what shape it takes, must have inner angles that sum up to 540°: S=(n-2)×180° S=(5-2)×180° S=3180° S=540° S=3180° S=540° S=3180° S=540° As a result, subtracting the four known angles from 540° will result in the following angle being left: 540°-105°-115°-109°-111°=100° The angle that is unknown is 100°.

## Finding Interior Angles of Regular Polygons

Once you’ve learned how to calculate the sum of interior angles of a polygon, obtaining one internal angle for any regular polygon is as simple as splitting the polygon in half. The following is the formula for a regular polygon, where S= the sum of the interior angles and n= the number of congruent sides: An octagon is shown here (eight sides, eight interior angles). To begin, apply the following formula to get the sum of internal angles: S=(n-2)180° S=(8-2)180° S=6180° S=1,080° S=(n-2)180° S=(8-2)180° S=6180° S=1,080° After that, divide the total by the number of sides to get the following result:

• The measure of each internal angle is equal to Sn
• The measure of each interior angle is equal to 1,080°8
• The measure of each interior angle is equal to 135°.

Every internal angle of a regular octagon is equal to 135 degrees.

## Finding the Number of Sides of a Polygon

If you know the value ofS, the sum of interior angles, you can apply the same method, S=(n-2)180°, to find out how many sides a polygon has if you know the value ofS. Even though you are aware that the total of the internal angles is 900°, you have no notion what the form is. To figure out what you don’t know, use the following formula to combine what you already know: Give the formula in full: S=(n-2)×180° Make use of what you already know, S=900° 900°=(n-2)180° 180 degrees should be divided between both sides.

5=n-2 Add 2 to both sides of the equation.

## Lesson Summary

In this stage, you should be able to recognize the interior angles of polygons, and you should be able to recall and use the formula S=(n-2)180° to get the total sum of all the interior angles of a polygon. Not only that, but you can also calculate the interior angles of polygons using Sn, and you can determine the number of sides of a polygon if you know the sum of its interior angles. That is a tremendous amount of knowledge derived from a single formula, S=(n-2)180°.

### Next Lesson:

Triangle’s midpoint is represented by the symbol

## 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°.

Try this with a square: 90° + 90° + 90° + 90° = 360°A square adds up to 360 degreesNow tilt a line by 10°: 80° + 100° + 90° + 90° = 360°It still adds up to 360 degreesThe Interior Angles of a Quadrilateral add up to 360 degreesThe Interior Angles of a Quadrilateral add 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.”

## How to find an angle in a polygon – SAT Math

Assuming that angles A and C are complementaries whereas angles B and D are supplementary angles, which of the following statements is correct? Answers that could be given: None of the options. BC = A/DB/CA * CB * DAD = A/DB/CA The correct response is: none of the options. Explanation: This question is extremely deceptive because, while each answer has the potential to be correct, none of them is required to be correct. There might be a very small angle (0.001 degrees) between angle A and angle C, and another angle that is extremely huge (0.001 degrees).

• On the other hand, the two perspectives may be quite similar.
• We can refute AD = BC by using the same data as above, which is 8100.009.
• The measure of angleA in the isosceles triangleABC is 50 degrees in length.
• Multiple possible solutions exist: there is more than one valid solution.
• The temperature is 80 degrees.
• The correct answer is 95 degrees.
• Explanation: Because 50 plus 50 plus x equals 180, and x equals 80, the vertex angle must be 80 degrees.

As a result, the two base angles must both be 65 degrees since 50 + x + x= 180 equals x= 65.

When constructing the triangle ABC, the measure of angle A equals 70 degrees, the measure of angle B equals x degrees, and the measure of angle C equals y degrees When y is expressed in terms of x, what is its value?

Take 70 away from both sides and you’ll notice that x+y=110.

Is there a way to find out how many degrees each interior angle of a normal convex polygon with twelve sides is in degrees?

Because the problem involves a polygon with twelve sides, we shall refer to it as n= 12.

As a result, all of the angles have the same measure since the polygon is regular (meaning that all of its sides are congruent).

For the simple answer, we must divide 1800 by 12, which results in 150.

PolygonABDFHGECis a regular octagon as seen in the diagram above.

Explanation: It is important to note that angleFHI is a supplement to angleFHG, which is an inside angle in the octagon.

As long as we can figure out what the measure of each internal angle in the octagon is, we can figure out what the supplement of angleFHG is, and that will give us the measure of angleFHI.

With eight sides, an octagon’s angles add up to 180(8)2 = 180(6) = 1080 degrees.

Its regularity ensures that all of the octagon’s sides and angles are consistent with one another.

Each angle in the polygon has a measure of 1080/8 = 135 degrees, which is the result of this calculation.

After determining the measure of angleFHG, we may determine the measure of angleFHI.

FHG and FHI We may formulate the following equation as a function of time: Measuring FHG plus measuring FHI equals 180135 + measuring FHI equals 180 Subtract 135 from both sides of the equation.

The correct response is: Explanation: An octagon is made up of six triangles, which equals 1080 degrees.

An octagon has eight central angles, each of which has a different measure.

Because there are 360 degrees and 8 angles, dividing leaves a 45-degree angle for each angle.

The correct response is: Explanation: The sum of the four angles of a quadrilateral is 360 degrees.

The sum of the six angles of a hexagon is 720.

The correct response is: Explanation: To solve, apply the formula for the total degrees in a polygon, where n is the number of vertices, and multiply the result by 100.

The correct response is: In the following example, if one exterior angle is measured at each vertex of any convex polygon, the total of their measurements equals Each exterior angle of a regular polygon – a polygon with congruent sides and congruent interior angles – is congruent to the angles of the interior angles of the polygon.

It is a full number if we take the common measure, multiply both sides by:, and divide the result by the number of sides in the common measure.

A simple check verifies that 360 divided by 8, 10, 12, or 15 produces a complete result, indicating that this is the proper decision.

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## Formula for Exterior Angles and Interior Angles, illustrated examples with practice problems on how to calculate.

When there aren sides to a convex polygon, the sum of the measurements of the interioranglesis\$ (n-2)cdot180\$ (n-2)cdot180\$

Shape Formula Sum Interior Angles
\$\$ red 3\$\$ sided polygon(triangle) \$\$ (red 3-2) cdot180\$\$ \$\$ 180^\$\$
\$\$ red 4\$\$ sided polygon(quadrilateral) \$\$ (red 4-2) cdot 180 \$\$ \$\$ 360^\$\$
\$\$ red 6 \$\$ sided polygon(hexagon) \$\$ (red 6-2) cdot 180 \$\$ \$\$ 720^\$\$
##### Problem 1

What is the total number of degrees that all of the interior angles of an atriangle have in common? 180° You may also make use of the Interior Angle Theorem, which is as follows: \$\$ (red 3 -2) cdot 180=(1) cdot 180 = 180 \$\$ (red 3 -2) cdot 180=(1) cdot 180 = 180 \$\$ (red 3 -2) cdot 180 = 180 \$\$ (red 3 -2) cdot 180 = 180 \$\$

##### Problem 2

Do you know how many degrees there are in total between all the internal angles of the polygon?

##### Problem 3

In a polygon (a pentagon), what is the sum measure of the interior angles on the sides?

##### Problem 4

Who knows how many internal angles there are in a polygon (a hexagon), but we do know how many there are.

### VideoTutorialon Interior Angles of a Polygon

Quite simply, a regular polygon is one whose sides are all of the same length and whose angles are all of the same measurement. You’ve probably heard of the right triangle and the equilateral triangle, which are the two most well-known and most often studied varieties of regular polygons in mathematics and geometry classes.

#### Examples of Regular Polygons

Hexagon in its regular form Pentagon operations on a regular basis More information on regular polygons may be found here.

### Measure of a Single Interior Angle

Shape Formula Sum interior Angles
Regular Pentagon \$\$ (red 3-2) cdot180\$\$ \$\$ 180^\$\$
\$\$ red 4\$\$ sided polygon(quadrilateral) \$\$ (red 4-2) cdot 180 \$\$ \$\$ 360^\$\$
\$\$ red 6 \$\$ sided polygon(hexagon) \$\$ (red 6-2) cdot 180 \$\$ \$\$ 720^\$\$

#### What about when you just want 1 interior angle?

In order to find the measure of a single interiorangleof an irregular polygon (a polygon with sides of equal length and angles of equal measure) with n sides, we first calculate the sum of interior angles or \$\$ (red n-2) cdot 180 \$\$ and then divide that sum by the number of sides or \$\$ red n\$\$. In order to find the measure of a single interiorangleof an irregular polygon (a polygon with sides of equal length and angles of equal The Formula for Success In a regularpolygon with \$\$ red n \$\$ sides, the measure of any interior angle is given by the equation text=frac.

#### Example 1

Now let’s take a look at a classic example that you’re probably already familiar with: the triangle \$\$triangle\$\$ Now, keep in mind that the new rule mentioned above only applies to regular polygons. As a result, the only form of triangle we could possibly be talking about is an equilateral triangle, such as the one seen below. The sum of the inner angles of a triangle measures \$\$ 180 \$\$, as you may already be aware of this fact. as well as the fact that, in the exceptional situation of an equilateral triangle, each angle measures precisely \$\$ 60 \$\$ in length and width \$text=fractextred 3 \$text=fractextred 3 \$text=fractextred 3 \$text=fractextred 3 \$text=fractextred 3 \$text=fractextred 3 \$text=fractextred 3 \$text=fractextred 3 \$text=fractextred 3 \$text=fractextred 3 \$text=fractextred 3 \$text=fractextred 3 \$text=fractextred 3 \$text=fractextred 3 \$text=fractextred 3 \$text=fractextred 3 \$ Consequently, our new formula for determining the measure of an angle in an irregular polygon is compatible with the laws for angles of triangles that we have learned in previous lectures.

Figure 1.

#### Example 2

To obtain the measure of an interior angle of a regular octagon with eight sides, use the following formula:\$text\$text=frac\$frac= 135\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$

### Finding 1 interior angle of a regular Polygon

Is there a way to find out the size of one internal angle of a normal octagon?

##### Problem 6

Calculate the length of one internal angle of a regular dodecagon (12-sided polygon) using the formula below.

##### Problem 7

Calculate the length of one internal angle of a regular hexadecagon (16-sided polygon) using the formula below.

##### Challenge Problem

What is the length of the internal angle of a pentagon when it is one? This question cannot be answered because the shape is not a regular polygon as defined by the question format. If the polygon is not regular, you will only be able to apply the algorithm to identify a single internal angle. Consider, for example, their regular pentagon, which is shown below. Just by glancing at the picture, you can see that \$\$angle A and \$\$angle B \$\$are not congruent with one another. The lesson of this story is that while you may use our method to get the total of the interior angles of any polygon (regular or not), you cannot use the formula on this page to measure a single angle, unless the polygon is regular.

#### How about the measure of an exterior angle?

Angles on the outside of a polygon Formula for the sum of exterior angles:The sum of the measurements of the exterior angles of a polygon, one at each vertex, equals 360° when the polygon has six vertices.

### Measure of a Single Exterior Angle

Formula to determine one angle of a regularconvex polygon with n sides = \$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$ \$\$\$\$ \$\$\$\$ \$\$\$\$ \$\$\$\$ \$\$\$\$ \$\$\$\$ \$\$\$\$ \$\$\$\$ \$\$\$\$ \$\$\$\$ \$\$\$\$ \$\$\$\$ \$\$\$\$ \$\$\$\$ \$\$\$\$ \$\$\$\$ \$\$\$\$ \$\$\$\$ \$\$\$\$ \$\$\$\$ \$\$\$\$ \$\$\$\$ \$\$\$\$ \$\$\$\$ \$\$\$\$ \$\$\$\$ \$\$\$\$ \$\$\$\$ \$

### PracticeProblems

In a regular pentagon, what is the measure of one of its outer angles?

##### Problem 9

What is the length of one of the outside angles of a regulardecagon (a polygon with ten sides)?

##### Problem 10

1 external angle of a regular dodecagon (12-sided polygon) is measured in what unit of measure?

##### Challenge Problem

What is the length of one of the pentagon’s outside angles in meters? This question cannot be answered because the shape is not a regular polygon as defined by the question format. Despite the fact that you are aware that the sum of the outside angles is 360, you can only apply the method to obtain a single exterior angle if the polygon is regular in shape! Take, for example, the pentagon seen in the image below. Despite the fact that we know that all of the external angles add up to 360 °, we can tell by just looking that each\$\$ angle A textandangle B \$\$are not congruent with one another.

### Determine Number of Sides from Angles

It is feasible to determine the number of sides a polygon has depending on the number of degrees present in either its exterior or interior angles.

##### Problem 11

In this polygon, how many sides does it have if each outer angle measures 10 degrees?

##### Problem 12

How many sides does this polygon have if each of its outer angles is 10°?

##### Problem 13

How many sides does this polygon have if each outer angle measures 10°?

##### Challenge Problem

In this polygon, how many sides does it have if each outer angle measures 80 degrees? There is no satisfactory answer to this question. When you apply the method to obtain a single exterior angle in order to solve for the number of sides, you get a decimal (4.5), which is mathematically impossible to compute.

Consider the following: How is it possible for a polygon to have 4.5 sides? Aquadrilateral is a quadrilateral with four corners. A pentagon is a shape with five sides.

## SOLVED:Find the measure of each interior angle of quadrilateral R S T V.

Okay, so this problem has a quadrilateral in it, don’t you think so? Make an arrest TV show with inside views of being at five in, or Auntie will get two in and an ass from the outside. Is it six o’clock already? A quadrilateral whose internal angles add up to a total of 360 degrees. So the total has to be 360, right? That was obtained by multiplying four by two, which equals two times one hundred eighty. For quadrilateral, there are certain formulas to follow, and then three sixes is something you should be able to recall quickly.

• Let’s see, if we add up all the numbers on the ends, we get 2 plus 6 + 5 plus 2 = 12.
• If we divide both sides equally, we’ll have roughly 15 in total, which is equivalent to 24.
• It’s that simple.
• It’s only twice a day for 24 hours.
• So it’s going to be 100 degrees and 20 degrees, and then it’ll be six times 24 degrees.
• Okay, thank you so much for your assistance.

## Interior Angles of a Polygon (13 Step-by-Step Examples!)

Do you have trouble figuring out how to get the internal angles of a polygon? In this geometry lesson, you will learn just that. Jenn, Founder Calcworkshop ®, 15+ Years Experience (LicensedCertified Teacher)You have come to the correct spot because that is exactly what you will learn in today’s geometry lesson. Let’s get started right away!

## Interior Angles

What if I told you that triangles play a key role in determining how to aggregate all of the inner angles of any convex polygon? Recall that a convex polygon does not have any angles that point inward, but a concave polygon creates something that appears to be a cave, with angles pointing inward into the interior of the polygon. Regular polygons are also those that are both equilateral (all sides are congruent) and equiangular (all angles are congruent) (all angles are congruent). Let’s start with a polygon of any size and draw diagonals from one vertex to the other.

• And the total of the internal angles is determined by the number of triangles that may be constructed.
• How?
• As Math is Funnicely points out, this means that every time we add a side, we are adding another 180° to the total, which is what we want.
• (where n is the number of sides on which the polygon is formed) The formula for each of the most frequent polygons is depicted in the chart to the right (triangle, quadrilateral, pentagon, hexagon, etc.).

Polygon Chart You’ll discover how to achieve this by watching the video below, which outlines the steps to take. Formula for the Interior Angle

## Exterior Angles

What’s more, did you know that when you have one angle at each vertex, the total of the measurements of the outer angles equals 360 degrees? As a result, if we have a regular polygon, the measure of each outside angle is equal to 360°/n degrees. In the case of a regular pentagon (5-sided polygon with equal angles and sides), each exterior angle is equal to the quotient of 360 degrees and the number of sides as stated in the table below. Angles on the outside are added together. You’ll learn how to do the following things in the video below:

1. Calculate the sum of internal angles for a variety of polygonal shapes. To find out the size of each inner and exterior angle of a regular polygon, use this formula. You are given the measure of one exterior or interior angle of a regular polygon
2. You must determine the number of sides the polygon possesses. Using our new formulae and properties, we can figure out the measurements of unknown angles in a polygon

## Video – LessonExamples

• Angles of Polygons: An Introduction to the Video
• In this section, we will discuss formulas for calculating the total angle sum of a polygon’s inner and exterior angles. Members-Only Content
• Exclusive Content for Members Only
• Experiment with regular polygons to find the total of the interior angles and the measure of each interior and exterior angle (Examples 1–5). 00:12:01– Calculate the sum of the interior angles and the measure of each interior and exterior angle (Examples 1–5). Experiment with the number of sides of a regular polygon given an exterior angle (Examples 6-8)
• 00:23:37– Determine the number of sides of a regular polygon given an external angle
• 00:26:57– Given an interior angle of a regular polygon, determine the number of sides (Examples 9-11)
• 00:26:57– Given an interior angle of a regular polygon, find the number of sides (Examples 9-11)
• A method for finding the interior angles of an irregular polygon (Example12)
• 00:33:54– A method for finding the internal angles of an irregular polygon
• In the provided polygon (Example13), 00:38:04 is the time to find all of the unknown angles. Questions and solutions for practice problems with detailed step-by-step solutions
• Chapter tests with video solutions
See also:  How Does The Sodium-Potassium Pump Make The Interior Of The Cell Negatively Charged

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## Interior Angles of a Polygon

Let’s go over a few crucial words to make sure we’re all on the same page before we continue. Please keep in mind that an apolygon is a two-dimensional object whose sides are drawn with straight lines (no curves), and which when joined together forms a closed space. Avertex is a point on a polygon where two sides come together and form a triangle. There is an internal angle of the polygon at each of its vertices. As an example, a square contains four inner angles, each of which measures 90 degrees.

### Sum of the interior angles

If we go even further, if the polygon has x sides, then the sum (S) of the degree measures of the x interior sides is given by the formula S = (x – 2) where x is the number of sides (180). For example, a triangle has three angles that total up to 180 degrees when added together. A square has four angles that add up to a total of 360 degrees. It is necessary to double the overall amount by 180 degrees for each new side that is added on. For the time being, let us consider the diagonal. In any case, what exactly is adiagonal?

If you exclude the lines that are also sides of the polygon, it is made up of all the lines connecting the points in the polygon.

The line segment BD divides the quadrilateral ABCD into two triangles, as seen in the diagram.

## Example 1

The quadrilateral ABCD has four angles, as you might expect. 2:3:3:4 are the angles formed by those four points. The degree measure of the largest angle in the quadrilateral ABCD has to be determined.

### What do we know?

In this case, we have four unknown angles, as well as knowledge regarding their relationship to one another. Considering that we already know that the total of all four angles must equal 360 degrees, all we need is an equation that sums our four unknown angles and sets them all to 360 degrees.

Because they are in a ratio, they must have some common element, which we will name x, which we will need to identify.

### Steps:

1. Add the words 2x + 3x + 3x + 4x to make a total of 4x. Calculate the total of the terms to be 360 degrees
2. Calculate the value of x
3. Calculate the angle in degrees by using the formula below.

### Solve

360 is equal to 2x + 3x + 3x + 4x. 12x = 360 x = 360/12 x = 30 12x = 360 x = 360/12 x = 30 Despite the fact that we know x = 30, we are not finished yet. In order to determine the largest angle, we multiply 30 times 4. Because 30 multiplied by 4 equals 120, the largest angle is 120 degrees. Similarly, the other angles are as follows: 3 * 30 = 90, 3 * 30 = 90, and 2 * 30 = 60.

### Regular Polygons

Equiangular polygons are found in regular polygons. Every one of its angles has the same measurement. In addition, it is equilateral. Each of its sides is the same length as the other. Squares are regular polygons, and while a square is a form of rectangle, rectangular shapes that are not squares are not regular polygons, and hence are not regular polygons.

## Example 2

Find the sum of the degree measurements of the angles of a hexagon by multiplying them together. Find each internal angle’s degree measure based on the assumption that the hexagon is regular.

### What do we know?

To sum the degree measure of any polygon, we may apply the formula S = (x – 2)(180) as shown below. Because a hexagon has six sides, the number x equals six.

### Solve

Let x = 6 be used in the formula, and simplify it as follows: S is equal to (6 – 2) (180) S is equal to four (180) S = 720 Amperes It is known as regular polygonisequiangular because all of its angles have the same measure. In the case of a normal hexagon, the sum of 720 degrees would be spread evenly across the six sides of the hexagonal shape. As a result, 720/6 = 120. In a regular hexagon, there are six angles, each of which is 120 degrees in length.

## Example 3

In a polygon with 3600 degrees of angles, how many sides does it have?

### Reversing the formula

We may utilize the formula S = (x – 2)(180) to solve for x once more, but this time we’ll be solving for x instead of S. It’s not a huge deal!

### Solve

In this problem, let S = 3600 be the starting point and solve for x. 3600 is equal to (x – 2) (180) 3600 is equal to 180x – 360. 3600 + 360 = 180x 3960 = 180x 3960/180 = x 22 = xA polygon with 22 sides has 22 angles whose sum is 3600 degrees. 3600 + 360 = 180x 3960 = 180x 3960/180 = x 22 = xA polygon with 22 sides has 22 angles whose sum is 3600 degrees.

### Exterior Angles of a Polygon

An external angle may be generated at each vertex of a polygon by extending one side of the polygon so that the interior and exterior angles at that vertex are supplementary (add up to 180). The angles a, b, c, and d in the diagram below are all outside, and the total of their degree measurements is 360 degrees. If a regular polygon has x sides, the degree measure of each exterior angle is equal to 360 divided by the number of sides in the polygon. Let’s have a look at two examples of test questions.

## Example 4

Calculate the degree measure of each inner and exterior angle of a regular hexagon by using the formula below. Consider the formula S=(x-2)*180, which denotes the sum of the internal angles of a triangle. A hexagon is made up of six sides. Since x = 6, the total S may be calculated by dividing x by two and multiplying the result by six (180) S is equal to (10 -6) (180) S is equal to four (180) S is equal to 720. In a hexagon, there are six angles, and in a regular hexagon, they are all of the same size.

Each is 720/6, or 120 degrees in length and width. Considering that inner and exterior angles are complementary (that is, they sum up to 180 degrees) at each vertex, the measure of each external angle is 180 minus 120 degrees, or 60.

## Example 5

In a regular polygon with 150 sides, the measure of each interior angle is 150. Determine the number of sides in the polygon. Previously, we determined the number of sides of a polygon by adding all the angles and solving the problem using the S=(x-2)*180 method. However, we just know the length of each inner angle this time around. To get the total number of angles, we’d have to multiply by the number of angles. However, the entire difficulty is that neither the number of sides nor the total are known at this time.

This is due to the fact that they are additional pairs (interior+exterior=180).

We now have a method for locating the solution!

By the way, a dodecagon is a geometric form with 12 sides that may be found in mathematics.

Feliz taught us a valuable lesson.