How To Find The Measure Of Each Interior Angle

Interior Angles of A Polygon: In Mathematics, an angle is defined as the effigy formed by joining the ii 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 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 mostly measured using degrees or radians. And so, if a polygon has 4 sides, then it has four angles besides. 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 bending theorem
Outside angles of Polygon
Solved Examples
FAQs

What is Meant past Interior Angles of a Polygon?

An interior angle of a polygon is an bending formed within the two next sides of a polygon. Or, we can say that the bending measures at the interior function of a polygon are called the interior angle of a polygon. Nosotros know that the polygon tin be classified into two dissimilar types, namely:

Regular Polygon
Irregular Polygon

For a regular polygon, all the interior angles are of the same measure out. But for irregular polygon, each interior angle may have different measurements.

Sum of Interior Angles of a Polygon

The Sum of interior angles of a polygon is always a abiding value. If the polygon is regular or irregular, the sum of its interior angles remains the same. 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-two) degrees

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

Polygon Proper name	Number of Interior Angles	Sum of Interior 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 three sides and three angles. Since, we know, there is a total of iii types of triangles based on sides and angles. Just the angle of the sum of all the types of interior angles is always equal to 180 degrees. For a regular triangle, each interior bending will be equal to:

180/3 = threescore degrees

sixty°+60°+sixty° = 180°

Therefore, no thing if the triangle is an acute triangle or birdbrained 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 up across different types of quadrilaterals, such as:

Square
Rectangle
Parallelogram
Rhomb
Trapezium
Kite

All the shapes listed above have iv sides and four angles. The common property for all the in a higher place iv-sided shapes is the sum of interior angles is ever equal to 360 degrees. For a regular quadrilateral such as foursquare, each interior angle will be equal to:

360/four = 90 degrees.

xc° + 90° + 90° + ninety° = 360°

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

Interior angles of Pentagon

In case of the pentagon, it has v sides and also information technology can be formed past joining three triangles side past side. Thus, if one triangle has sum of angles equal to 180 degrees, therefore, the sum of angles of three triangles volition be:

3 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 angle
Triangle	60°
Quadrilateral	90°
Pentagon	108°
Hexagon	120°
Septagon	128.57°
Octagon	135°
Nonagon	140°
Decagon	144°

Interior Bending Formulas

The interior angles of a polygon always lie inside the polygon. The formula tin can be obtained in three ways. Allow the states talk over the three unlike formulas in detail.

Method 1:

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

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

Method ii:

If the outside angle of a polygon is given, and then the formula to find the interior angle is

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

Method 3:

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

Interior Angle = Sum of the interior angles of a polygon / due north

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 – 4) × ninety°.

To prove:

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

Proof:

Interior angles example

ABCDE is a "n" sided polygon. Accept whatsoever signal O within the polygon. Join OA, OB, OC.

For "north" 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 above value in (1), we get

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

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

Take 90 as common, then it becomes

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

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

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

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

Exterior 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 side by side interior angle and exterior bending for any polygon is equal to 180 degrees since they course a linear pair. Also, the sum of outside angles of a polygon is ever equal to 360 degrees.

Outside angle of a polygon = 360 ÷ number of sides

Exterior Angles of a Polygon
Exterior 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 have?

Solution:

Given: Each interior angle = 144°

We know that,

Interior angle + Outside angle = 180°

Exterior angle = 180°-144°

Therefore, the exterior angle is 36°

The formula to observe 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.ii: What is the value of the interior angle of a regular octagon?

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

n = 8

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

Sum = (eight-two) 10 180° = 6 x 180° = 1080°

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

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

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

Answer: Given,

Number of sides, due north = 10

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

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?

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

How to discover the sum of interior angles by the angle sum holding 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, there can be iv triangles that tin be formed. Thus,
4 x 180° = 720 degrees.

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

A decagon has 10 sides and 10 angles.
Sum of interior angles = (x – 2) x 180°
= eight × 180°
= 1440°
A regular decagon has all its interior angles equal in mensurate. 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 will exist 360°.

Source: https://byjus.com/maths/interior-angles-of-a-polygon/

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