9 Sided Polygon Interior Angles

Interior Angles of A Polygon: In Mathematics, an angle is defined as the effigy formed by joining the 2 rays at the common endpoint. An interior angle is an bending within a shape. The polygons are the airtight shape that has sides and vertices. A regular polygon has all its interior angles equal to each other. For case, a foursquare has all its interior angles equal to the right angle or 90 degrees.

The interior angles of a polygon are equal to a number of sides. Angles are generally measured using degrees or radians. So, if a polygon has 4 sides, and so it has 4 angles also. Also, the sum of interior angles of different polygons is different.

Table of Contents:

Definition
Sum of interior angles
- Interior angles of triangle
- Interior angles of quadrilateral
- Interior angles of pentagon
- Interior angles of regular polygon
Formulas
Interior angle theorem
Exterior angles of Polygon
Solved Examples
FAQs

What is Meant by Interior Angles of a Polygon?

An interior angle of a polygon is an angle formed inside the 2 adjacent sides of a polygon. Or, nosotros can say that the angle measures at the interior part of a polygon are called the interior angle of a polygon. We know that the polygon tin can be classified into two different types, namely:

Regular Polygon
Irregular Polygon

For a regular polygon, all the interior angles are of the aforementioned measure out. Merely for irregular polygon, each interior bending may take unlike measurements.

Sum of Interior Angles of a Polygon

The Sum of interior angles of a polygon is always a constant value. If the polygon is regular or irregular, the sum of its interior angles remains the aforementioned. Therefore, the sum of the interior angles of the polygon is given past the formula:

Sum of the Interior Angles of a Polygon = 180 (due north-2) degrees

As we know, there are unlike types of polygons. Therefore, the number of interior angles and the respective sum of angles is given beneath in the table.

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

Interior angles of Triangles

A triangle is a polygon that has three sides and 3 angles. Since, we know, at that place is a total of three types of triangles based on sides and angles. But the angle of the sum of all the types of interior angles is e'er equal to 180 degrees. For a regular triangle, each interior angle will exist equal to:

180/3 = 60 degrees

60°+sixty°+60° = 180°

Therefore, no matter if the triangle is an acute triangle or obtuse triangle or a right triangle, the sum of all its interior angles will always be 180 degrees.

Interior Angles of Quadrilaterals

In geometry, we have come beyond different types of quadrilaterals, such as:

Square
Rectangle
Parallelogram
Rhombus
Trapezium
Kite

All the shapes listed above accept four sides and iv angles. The common property for all the above four-sided shapes is the sum of interior angles is ever equal to 360 degrees. For a regular quadrilateral such every bit square, each interior angle will be equal to:

360/four = 90 degrees.

ninety° + xc° + xc° + 90° = 360°

Since each quadrilateral is made up of two triangles, therefore the sum of interior angles of two triangles is equal to 360 degrees and hence for the quadrilateral.

Interior angles of Pentagon

In instance of the pentagon, it has five sides and besides information technology can be formed by joining three triangles side by side. Thus, if one triangle has sum of angles equal to 180 degrees, therefore, the sum of angles of three triangles volition exist:

iii x 180 = 540 degrees

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

For a regular pentagon, each angle volition be equal to:

540°/5 = 108°

108°+108°+108°+108°+108° = 540°

Sum of Interior angles of a Polygon = (Number of triangles formed in the polygon) x 180°

Interior angles of Regular Polygons

A regular polygon has all its angles equal in measure.

Regular Polygon Name	Each interior bending
Triangle	sixty°
Quadrilateral	90°
Pentagon	108°
Hexagon	120°
Septagon	128.57°
Octagon	135°
Nonagon	140°
Decagon	144°

Interior Angle Formulas

The interior angles of a polygon ever prevarication inside the polygon. The formula can be obtained in three means. Let the states discuss the three different formulas in detail.

Method i:

If "n" is the number of sides of a polygon, then the formula is given below:

Interior angles of a Regular Polygon = [180°(n) – 360°] / due north

Method two:

If the exterior bending of a polygon is given, then the formula to detect the interior bending is

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

Method 3:

If nosotros know the sum of all the interior angles of a regular polygon, we can obtain the interior angle by dividing the sum by the number of sides.

Interior Angle = Sum of the interior angles of a polygon / n

Where

"n" is the number of polygon sides.

Interior Angles Theorem

Below is the proof for the polygon interior bending sum theorem

Statement:

In a polygon of 'north' sides, the sum of the interior angles is equal to (2n – 4) × 90°.

To bear witness:

The sum of the interior angles = (2n – 4) right angles

Proof:

Interior angles example

ABCDE is a "n" sided polygon. Have any bespeak O inside the polygon. Join OA, OB, OC.

For "northward" sided polygon, the polygon forms "n" triangles.

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

Therefore, the sum of the angles of n triangles = n × 180°

From the above statement, we can say that

Sum of interior angles + Sum of the angles at O = 2n × ninety° ——(one)

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

Substitute the to a higher place value in (1), we get

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

Then, the sum of the interior angles = (2n × 90°) – 360°

Have 90 as mutual, so it becomes

The sum of the interior angles = (2n – 4) × xc°

Therefore, the sum of "n" interior angles is (2n – 4) × xc°

So, each interior angle of a regular polygon is [(2n – 4) × 90°] / northward

Note: In a regular polygon, all the interior angles are of the same measure out.

Exterior Angles

Exterior angles of a polygon are the angles at the vertices of the polygon, that prevarication outside the shape. The angles are formed by one side of the polygon and extension of the other side. The sum of an side by side interior angle and exterior bending for any polygon is equal to 180 degrees since they form a linear pair. As well, the sum of exterior angles of a polygon is ever equal to 360 degrees.

Exterior angle of a polygon = 360 ÷ number of sides

Exterior Angles of a Polygon
Exterior Angle Theorem
Alternating Interior Angles
Polygon

Solved Examples

Q.i: If each interior bending is equal to 144°, so how many sides does a regular polygon have?

Solution:

Given: Each interior angle = 144°

We know that,

Interior angle + Exterior angle = 180°

Exterior bending = 180°-144°

Therefore, the exterior bending is 36°

The formula to find the number of sides of a regular polygon is as follows:

Number of Sides of a Regular Polygon = 360° / Magnitude of each exterior bending

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

Hence, the polygon has 10 sides.

Q.two: What is the value of the interior angle of a regular octagon?

Solution: A regular octagon has viii sides and eight angles.

due north = 8

Since, nosotros know that, the sum of interior angles of octagon, is;

Sum = (viii-2) x 180° = half dozen x 180° = 1080°

A regular octagon has all its interior angles equal in mensurate.

Therefore, measure of each interior bending = 1080°/8 = 135°.

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

Answer: Given,

Number of sides, n = x

Sum of interior angles = (x – two) 10 180° = viii ten 180° = 1440°.

Video Lesson on Angle sum and exterior angle holding

Practise Questions

Find the number of sides of a polygon, if each bending is equal to 135 degrees.
What is the sum of interior angles of a nonagon?

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Frequently Asked Questions – FAQs

What are the interior angles of a polygon?

Interior angles of a polygon are the angles that lie at the vertices, inside the polygon.

What is the formula to find the sum of interior angles of a polygon?

To observe the sum of interior angles of a polygon, use the given formula:
Sum = (northward-2) x 180°
Where n is the number of sides or number of angles of polygons.

How to notice the sum of interior angles by the angle sum property of the triangle?

To detect the sum of interior angles of a polygon, multiply the number of triangles formed inside the polygon to 180 degrees. For instance, in a hexagon, there can be four triangles that tin can be formed. Thus,
4 ten 180° = 720 degrees.

What is the measure of each bending of a regular decagon?

A decagon has x sides and 10 angles.
Sum of interior angles = (10 – 2) x 180°
= 8 × 180°
= 1440°
A regular decagon has all its interior angles equal in measure. Therefore,
Each interior angle of decagon = 1440°/10 = 144°

What is the sum of interior angles of a kite?

A kite is a quadrilateral. Therefore, the angle sum of a kite will exist 360°.