Interior angles of A Polygon: In Mathematics, an angle is defined as the number formed by involvement the 2 rays in ~ the typical endpoint. An internal angle is an angle inside a shape. The polygons room the closed form that has sides and vertices. A continual polygon has actually all its internal angles same to every other. Because that example, a square has all its internal angles equal to the ideal angle or 90 degrees.

The interior angles that a polygon are equal come a number of sides. Angle are typically measured using levels or radians. So, if a polygon has actually 4 sides, climate it has four angles together well. Also, the sum of inner angles of various polygons is different.

 Table that Contents:Sum of inner angles

## What is supposed by interior Angles the a Polygon?

An inner angle of a polygon is one angle formed inside the two surrounding sides that a polygon. Or, we deserve to say that the angle steps at the interior part of a polygon are referred to as the interior angle that a polygon. We know that the polygon can be classified into two different types, namely:

Regular PolygonIrregular Polygon

For a continuous polygon, all the interior angles space of the exact same measure. But for irregular polygon, each inner angle might have different measurements.

You are watching: What is the measure of each interior angle of a regular octagon?

## Sum of internal Angles the a Polygon

The amount of inner angles that a polygon is constantly a continuous value. If the polygon is consistent or irregular, the sum of its inner angles remains the same. Therefore, the sum of the internal angles the the polygon is provided by the formula:

Sum of the inner Angles of a Polygon = 180 (n-2) degrees

As us know, there are different varieties of polygons. Therefore, the variety of interior angles and the particular sum of angle is given listed below in the table.

 Polygon Name Number of internal Angles Sum of inner Angles = (n-2) x 180° Triangle 3 180° Quadrilateral 4 360° Pentagon 5 540° Hexagon 6 720° Septagon 7 900° Octagon 8 1080° Nonagon 9 1260° Decagon 10 1440°

### Interior angles of Triangles

A triangle is a polygon that has actually three sides and three angles. Since, us know, there is a full of three varieties of triangles based on sides and angles. Yet the angle of the sum of every the varieties of interior angles is always equal to 180 degrees. Because that a constant triangle, each interior angle will certainly be equal to:

180/3 = 60 degrees

60°+60°+60° = 180°

Therefore, no matter if the triangle is one acute triangle or obtuse triangle or a right triangle, the sum of all its internal angles will constantly be 180 degrees.

In geometry, we have actually come throughout different species of quadrilaterals, together as:

SquareRectangleParallelogramRhombusTrapeziumKite

All the shapes detailed above have four sides and also four angles. The common property for every the over four-sided shapes is the sum of internal angles is always equal come 360 degrees. Because that a continual quadrilateral such as square, each interior angle will certainly be same to:

360/4 = 90 degrees.

90° + 90° + 90° + 90° = 360°

Since each quadrilateral is consisted of of 2 triangles, thus the amount of interior angles of 2 triangles is equal to 360 degrees and hence because that the quadrilateral.

### Interior angles of Pentagon

In situation of the pentagon, that has 5 sides and also it deserve to be developed by joining three triangles next by side. Thus, if one triangle has actually sum of angle equal to 180 degrees, therefore, the amount of angle of 3 triangles will certainly be:

3 x 180 = 540 degrees

Thus, the angle sum of the pentagon is 540 degrees.

For a regular pentagon, every angle will certainly be same to:

540°/5 = 108°

108°+108°+108°+108°+108° = 540°

 Sum of internal angles that a Polygon = (Number that triangles created in the polygon) x 180°

## Interior angles of continuous Polygons

A continual polygon has all its angles equal in measure.

 Regular Polygon Name Each inner angle Triangle 60° Quadrilateral 90° Pentagon 108° Hexagon 120° Septagon 128.57° Octagon 135° Nonagon 140° Decagon 144°

## Interior angle Formulas

The interior angles of a polygon always lie within the polygon. The formula deserve to be acquired in three ways. Allow us discuss the three different formulas in detail.

Method 1:

If “n” is the number of sides that a polygon, then the formula is offered below:

Interior angle of a continuous Polygon = <180°(n) – 360°> / n

Method 2:

If the exterior angle of a polygon is given, climate the formula to find the inner angle is

Interior edge of a polygon = 180° – Exterior angle of a polygon

Method 3:

If we know the sum of every the internal angles that a constant polygon, we can acquire the interior angle by splitting the sum by the number of sides.

Interior angle = sum of the interior angles that a polygon / n

Where

“n” is the number of polygon sides.

## Interior angle Theorem

Below is the proof because that the polygon inner angle amount theorem

Statement:

In a polygon the ‘n’ sides, the amount of the internal angles is same to (2n – 4) × 90°.

To prove:

The amount of the inner angles = (2n – 4) appropriate angles

Proof: ABCDE is a “n” face polygon. Take any allude O within the polygon. Join OA, OB, OC.

For “n” face polygon, the polygon develops “n” triangles.

We understand that the sum of the angles of a triangle is equal to 180 degrees

Therefore, the amount of the angle of n triangle = n × 180°

From the above statement, we have the right to say that

Sum of inner angles + sum of the angles at O = 2n × 90° ——(1)

But, the amount of the angles at O = 360°

Substitute the over value in (1), us get

Sum of interior angles + 360°= 2n × 90°

So, the sum of the inner angles = (2n × 90°) – 360°

Take 90 together common, then it becomes

The amount of the internal angles = (2n – 4) × 90°

Therefore, the sum of “n” interior angles is (2n – 4) × 90°

So, each internal angle of a regular polygon is <(2n – 4) × 90°> / n

Note: In a continual polygon, every the internal angles room of the same measure.

## Exterior Angles

Exterior angle of a polygon are the angle at the vertices that the polygon, the lie exterior the shape. The angle are formed by one next of the polygon and also extension that the other side. The amount of an surrounding interior angle and exterior angle for any type of polygon is same to 180 degrees due to the fact that they form a straight pair. Also, the amount of exterior angle of a polygon is always equal to 360 degrees.

## Solved Examples

Q.1: If each interior angle is same to 144°, climate how plenty of sides walk a consistent polygon have?

Solution:

Given: Each internal angle = 144°

We know that,

Interior angle + Exterior angle = 180°

Exterior edge = 180°-144°

Therefore, the exterior angle is 36°

The formula to uncover the number of sides that a continuous polygon is as follows:

Number of sides of a consistent Polygon = 360° / magnitude of each exterior angle

Therefore, the variety of sides = 360° / 36° = 10 sides

Hence, the polygon has actually 10 sides.

Q.2: What is the worth of the inner angle of a consistent octagon?

Solution: A continual octagon has eight sides and eight angles.

n = 8

Since, we recognize that, the sum of internal angles that octagon, is;

Sum = (8-2) x 180° = 6 x 180° = 1080°

A continuous octagon has actually all its inner angles same in measure.

Therefore, measure up of each internal angle = 1080°/8 = 135°.

Q.3: What is the sum of inner angles that a 10-sided polygon?