Polygon Interior Angle Sum Theorem

Interior Angles of A Polygon: In Mathematics, an angle is divers every bit the effigy formed by joining the 2 rays at the common endpoint. An interior bending is an angle inside a shape. The polygons are the closed shape that has sides and vertices. A regular polygon has all its interior angles equal to each other. For example, a square has all its interior angles equal to the correct bending or 90 degrees.

The interior angles of a polygon are equal to a number of sides. Angles are more often than not measured using degrees or radians. And then, if a polygon has 4 sides, then it has four angles too. Besides, the sum of interior angles of different polygons is unlike.

Tabular array 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 bending 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 chosen the interior angle of a polygon. We know that the polygon can be classified into 2 dissimilar types, namely:

Regular Polygon
Irregular Polygon

For a regular polygon, all the interior angles are of the same measure. Merely for irregular polygon, each interior angle may take different 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 by the formula:

Sum of the Interior Angles of a Polygon = 180 (n-ii) degrees

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

Polygon Name	Number of Interior Angles	Sum of Interior Angles = (n-2) x 180°
Triangle	3	180 °
Quadrilateral	iv	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 three sides and three angles. Since, we know, there 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 always equal to 180 degrees. For a regular triangle, each interior angle will exist equal to:

180/3 = 60 degrees

60°+60°+60° = 180°

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

Interior Angles of Quadrilaterals

In geometry, we accept come across different types of quadrilaterals, such as:

Square
Rectangle
Parallelogram
Rhombus
Trapezium
Kite

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

360/4 = ninety degrees.

90° + 90° + 90° + 90° = 360°

Since each quadrilateral is fabricated up of two triangles, therefore the sum of interior angles of 2 triangles is equal to 360 degrees and hence for the quadrilateral.

Interior angles of Pentagon

In example of the pentagon, it has five sides and also it tin can exist formed by joining three triangles adjacent. Thus, if one triangle has sum of angles equal to 180 degrees, therefore, the sum of angles of iii triangles will be:

3 x 180 = 540 degrees

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

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

540°/v = 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 mensurate.

Regular Polygon Proper noun	Each interior bending
Triangle	lx°
Quadrilateral	xc°
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 can be obtained in three ways. Permit us discuss the three different formulas in particular.

Method 1:

If "north" is the number of sides of a polygon, then the formula is given beneath:

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

Method 2:

If the exterior angle of a polygon is given, and so the formula to discover the interior angle is

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

Method iii:

If we know the sum of all the interior angles of a regular polygon, we tin can obtain the interior angle past 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 angle sum theorem

Statement:

In a polygon of 'northward' sides, the sum of the interior angles is equal to (2n – four) × ninety°.

To testify:

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

Proof:

Interior angles example

ABCDE is a "n" sided polygon. Take any point O within the polygon. Bring together OA, OB, OC.

For "n" sided polygon, the polygon forms "northward" 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 tin can say that

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

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

Substitute the higher up value in (1), we get

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

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

Accept 90 as common, then information technology becomes

The sum of the interior angles = (2n – iv) × 90°

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

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

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

Outside Angles

Exterior angles of a polygon are the angles at the vertices of the polygon, that lie outside the shape. The angles are formed by one side of the polygon and extension of the other side. The sum of an next interior angle and outside angle for whatever polygon is equal to 180 degrees since they form a linear pair. Also, the sum of exterior angles of a polygon is e'er equal to 360 degrees.

Exterior bending of a polygon = 360 ÷ number of sides

Related Manufactures

Exterior Angles of a Polygon
Outside Angle Theorem
Alternate Interior Angles
Polygon

Solved Examples

Q.1: If each interior angle is equal to 144°, then how many sides does a regular polygon take?

Solution:

Given: Each interior angle = 144°

We know that,

Interior bending + Exterior angle = 180°

Exterior angle = 180°-144°

Therefore, the exterior angle is 36°

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

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

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

Hence, the polygon has x sides.

Q.2: What is the value of the interior angle of a regular octagon?

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

n = viii

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

Sum = (8-ii) x 180° = vi x 180° = 1080°

A regular octagon has all its interior angles equal in measure out.

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

Q.iii: What is the sum of interior angles of a ten-sided polygon?

Answer: Given,

Number of sides, n = 10

Sum of interior angles = (10 – 2) x 180° = 8 x 180° = 1440°.

Video Lesson on Angle sum and exterior angle property

Do Questions

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

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Ofttimes 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 find the sum of interior angles of a polygon, use the given formula:
Sum = (north-2) 10 180°
Where n is the number of sides or number of angles of polygons.

How to notice the sum of interior angles by the bending sum belongings of the triangle?

To find 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, in that location can be four triangles that can be formed. Thus,
4 x 180° = 720 degrees.

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

A decagon has ten sides and 10 angles.
Sum of interior angles = (10 – 2) x 180°
= eight × 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 bending sum of a kite volition be 360°.