# How To Find The Measure Of An Interior Angle

## Exterior Angle of a Triangle

An exterior angle of a triangle can be generated at each vertex of a triangle by extending ONE SIDE of the triangle from the vertex to the base. Take a look at the image below.

### Calculating the Angles

To express the angles stated above, we may use equations to represent the measurements. One equation could be able to give us the sum of the angles of a triangle, for example. For example, the sum of x, y, and z equals 180. We know this to be true because the total of the angles inside a triangle is always 180 degrees, regardless of the shape of the triangle. What is the letter w? We haven’t figured it out yet. Due to the fact that they are a pair of supplementary angles, we can see that the measure of angle w plus the measure of angle Z equals 180 degrees.

That’s a 180-degree turn.

Here’s how you can go about it: The sum of x, y, and z equals 180 (this is the first equation) w plus z equals 180 (this is the second equation) Write the second equation as z = 180-w and substitute that value for the value of z in the first equation: z = 180-w x + y + (180 – w) = 180x + y – w = 0 x + y = w x + y = w x + y = w Interesting.

In fact, there is a theory known as theExterior Angle Theorem that goes into further detail about this relationship:

### Exterior Angle Theorem

It is known as our w in triangles. The measure of an exterior angle (our w) in a triangle is the sum of the measurements of the two remote interior angles (our x and y) in a triangle. Let’s take a look at two real-world challenges.

#### Example A:

The outside angle’s measure is (3x-10) degrees, while the two remote interior angles’ measures are 25 degrees and (x + 15) degrees, respectively. Find the value of x in this situation: In order to answer the problem, we will use the knowledge that W = X + Y. Please keep in mind that I’m referring to the angles W, X, and Y that can be seen in the first image of this lesson when I say “angles.” Their given names are not significant. What matters is that an exterior angle is equal to the sum of the distant interior angles on both sides of the line.

The outside angle equals the interior angle plus the other interior angle.

For this, we require the angles themselves, which can be computed as (3x-10), 25, and (x+15) respectively. The angles are 65 degrees, 25 degrees, and 40 degrees, respectively.

#### Example B

The external angle that has been specified is 110 degrees. Two distant interior angles have measurements of 50 and (2x + 30). Find the value of x. Keep in mind that exterior equals the sum of remote interior angles. We’re provided the perspective from the outside (110). We may deduce that 110 = (2x + 30) + 50 and then solve for x. The sum of two times plus thirty and fifty dollars is one hundred dollars. money money money money money money money money money money money money money money money money money money money money money money money money money money money money money money money money money money money money money money money money money money money money money money money money money money money money money money money money money money money money money money money money money money money money money money money money money money money money money \$\$ \$\$ 30 equals 2x \$\$ \$\$ 15 equals x \$\$ Mr.

Feliz taught us a valuable lesson.

## 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°.

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## 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. Moreover, you can recollect a method for determining an unknown interior angle of a polygon by subtracting the known interior angles from the estimated total, which you learned in geometry class. Not only that, but you can also use Sn to calculate the interior angles of polygons, and if you know the sum of the interior angles of a polygon’s sides, you can figure out how many sides the polygon has in total.

### Next Lesson:

Triangle’s midpoint is represented by the symbol

## Interior Angles Of A Triangle (video lessons, examples, step-by-step solutions)

Pages that are related Angles on the outside of a triangle Triangles Come in Many Shapes Angles in a triangle are defined as follows: The following picture illustrates that the sum of the angles of a triangle is 180° in the given situation.

Continue reading for additional examples and solutions on how to discover the missing angles in a triangle by using the steps below.

### Interior Angles

The angles within a triangle are referred to as the internal angles of the triangle.

### Properties of Interior Angles

• Whenever a triangle has three internal angles, the total of those angles is always 180°. Due to the fact that the inside angles sum up to 180°, every angle must be less than 180°.

### Find missing angles inside a triangle

For example, in the following triangle, determine the value of x. Solutions are as follows: x 24° + 32° = 180° (the total of the angles equals 180°). In this case, x + 56 degrees = 180 degrees; in this case, x = 180 degrees–56 degrees=124 degrees. In a triangle, how to locate the missing angle is explained. For example, in a triangle, find the angle that is missing. A = 15°, B =?, and C = 92° are the coordinates.

### Using Interior Angles Of Triangle To Set Up Equations

The sum of the internal angles of a triangle can be used to address difficulties in a variety of situations. Example:

1. Calculate the unknown by writing an equation and solving for it
2. Fill in the blanks with your response in each expression to find the measure of the angles
3. Provide justifications for your responses

In order to find the unknown, build an equation using the sum of the interior angles of a triangle and solve for the unknown. You can practice many arithmetic concepts by using the freeMathway calculator and problem solution provided below. Make use of the examples provided, or put in your own issue and cross-reference your answer with the step-by-step instructions. If you have any suggestions, comments, or concerns regarding this site or page, please let us know. Thank you for taking the time to provide comments or inquiries on ourFeedbackpage.

## 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.”

## 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 get the measure of an interior angle of a normal octagon with eight sides, use the following formula to apply the formula above as follows: \$text=text=fracfrac= 135\$ \$text=text=fracfrac= 135\$ \$text=text=fracfrac= 135\$ \$text=text=fracfrac= 135\$ \$text=text=fracfrac= 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.

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### 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.

## What Is the Formula for Finding Exterior and Interior Angles of a Polygon? [Solved]

In geometry, a polygon is a planar shape made by a limited number of line segments that are connected together.

## Answer: Interior Angles of a Regular Polygon with n sides: Interior angle = (180n – 360)/n

In order to solve this question, we will utilize the formula for the sum of internal angles and outside angles as our guide. Explanation: Inside angles are represented as the number of sides divided by 180 (n – 2), where n is the number of sides. Assuming that all interior angles of a regular polygon are equal, we can state that the interior angle of a polygon = sum of interior angles x number of sides= 180 (n-2)/2 n= (180n – 360)/n= (180n – 360)/n= (180n – 360)/n We now understand that the sum of the outside angles of a regular polygon is 360 degrees.

### So, interior Angles of a Regular Polygon with n sides is given by Interior angle = (180n – 360)/n and exterior Angles of a Regular Polygon with n sides is given by Exterior angle = 360 ° /n

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.

1. And the total of the internal angles is determined by the number of triangles that may be constructed.
2. How?
3. 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.
4. (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

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