What is the sum of the measures of the interior angles of a regular polygon if each exterior 90? Full

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An Interior Angle is an angle inside a shape

Nội dung chính

Another example:

Triangles

The Interior Angles of a Triangle add up to 180°

Let’s try a triangle:

90° + 60° + 30° = 180°

It works for this triangle

Now tilt a line by 10°:

80° + 70° + 30° = 180°

It still works!
One angle
went up by 10°,
and the other went down by 10°

Quadrilaterals (Squares, etc)

(A Quadrilateral has 4 straight sides)

Let’s try a square:

90° + 90° + 90° + 90° = 360°

A Square adds up to 360°

Now tilt a line by 10°:

80° + 100° + 90° + 90° = 360°

It still adds up to 360°

The Interior Angles of a Quadrilateral add up to 360°

Because there are 2 triangles in a square …

The interior angles in a triangle add up to 180° …

… and for the square they add up to 360° …

… because the square can be made from two triangles!

Pentagon

A pentagon has 5 sides, and can be made from three triangles, so you know what …

… its interior angles add up to 3 × 180° = 540°

And when it is regular (all angles the same), then each angle is 540° / 5 = 108°

(Exercise: make sure each triangle here adds up to 180°, and check that the pentagon’s interior angles add up to 540°)

The Interior Angles of a Pentagon add up to 540°

The General Rule

Each time we add
a side (triangle to quadrilateral, quadrilateral to pentagon, etc), we add another 180° to the total:

So the general rule is:

Sum of Interior Angles = (n−2) × 180°

Each Angle (of a Regular Polygon) = (n−2) × 180° / n

Perhaps an example will help:

Example: What about a Regular Decagon (10 sides) ?

Sum of Interior Angles = (n−2) × 180°

= (10−2) × 180°

= 8 × 180°

= 1440°

And for a Regular Decagon:

Each interior angle = 1440°/10 = 144°

Note: Interior Angles are
sometimes called “Internal Angles”

Interior Angles of A Polygon: In Mathematics, an angle is defined as the figure formed by joining the two rays the common endpoint. An interior angle 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 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, then it has four angles as well. Also, the sum of interior angles of different polygons is different.

Table of Contents:

Interior angles of triangleInterior angles of quadrilateralInterior angles of pentagonInterior angles of regular polygon

FormulasInterior angle theoremExterior angles of PolygonSolved ExamplesFAQs

What is Meant by Interior Angles of a Polygon?

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

Regular PolygonIrregular Polygon

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

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 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-2) 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
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 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 be 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 right triangle, the sum of all its interior angles will always be 180 degrees.

Interior Angles of Quadrilaterals

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

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

360/4 = 90
degrees.

90° + 90° + 90° + 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 case of the pentagon, it has five sides and also it 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 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°/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 Angle Formulas

The interior angles of a polygon always lie inside the polygon. The formula can be obtained in three ways. Let us discuss the three different 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°(n) – 360°] / n

Method 2:

If the exterior angle of a
polygon is given, 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, 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 angle sum theorem

Statement:

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

To prove:

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

Proof:

ABCDE is a
“n” sided polygon. Take any point O inside the polygon. Join OA, OB, OC.

For “n” 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 O = 2n × 90° ——(1)

But, the sum of the angles 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 – 4) × 90°

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

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

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

Exterior Angles

Exterior angles of a polygon are the angles 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 adjacent interior angle and exterior angle for any polygon is equal to 180 degrees since they form a linear pair. Also, the sum of exterior angles of a polygon is always
equal to 360 degrees.

Exterior angle of a polygon = 360 ÷ number of sides

Exterior Angles of a PolygonExterior Angle TheoremAlternate Interior AnglesPolygon

Solved Examples

Q1: 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 + Exterior angle = 180°

Exterior angle = 180°-144°

Therefore, the exterior angle 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 angle

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

Hence, the polygon
has 10 sides.

Q2: What is the value of the interior angle of a regular octagon?

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

n = 8

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

Sum = (8-2) x 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°.

Q3: What is the sum of interior angles of a 10-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

Practise Questions

Find 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?

Frequently Asked Questions – FAQs

What are the interior angles of a polygon?

Interior angles of a polygon are the angles that lie 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 n is the number of sides or number of angles of polygons.

How to find the sum of interior angles by the angle sum property 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 example, in a hexagon, there 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 10 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 be 360°.

What is the interior angle of the polygon with 90 sides?

Interior angles of Regular Polygons.

What is the exterior angle of a 90 Gon?

1 Answer. Each exterior angle measures to 4o .

What is the sum of the measures on the interior angles of a regular polygon if each exterior angle measure?

The exterior angle and the interior angle of any polygon are supplementary and must add up to 180o .

What is the sum of the measures of an interior and exterior angle of a regular pentagon?

Pentagon is formed from three triangles, so the sum of angles in a pentagon = 3 × 180° = 540°.
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