Angles Of A Regular Pentagon

Interior Angles of A Polygon: In Mathematics, an angle is defined every bit the figure formed by joining the ii rays at the mutual endpoint. An interior bending 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 example, a square has all its interior angles equal to the correct 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. And then, if a polygon has four sides, then information technology has four angles as well. 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 past Interior Angles of a Polygon?

An interior angle of a polygon is an angle formed inside the 2 next sides of a polygon. Or, we tin 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 exist classified into two different types, namely:

Regular Polygon
Irregular Polygon

For a regular polygon, all the interior angles are of the same measure. But for irregular polygon, each interior bending may accept dissimilar 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 nosotros know, there are dissimilar types of polygons. Therefore, the number of interior angles and the corresponding sum of angles is given below in the table.

Polygon Name	Number of Interior Angles	Sum of Interior Angles = (n-2) 10 180°
Triangle	3	180 °
Quadrilateral	four	360 °
Pentagon	five	540 °
Hexagon	six	720 °
Septagon	7	900 °
Octagon	8	1080 °
Nonagon	9	1260 °
Decagon	10	1440 °

Interior angles of Triangles

A triangle is a polygon that has iii sides and 3 angles. Since, we know, there is a total of three types of triangles based on sides and angles. But the bending of the sum of all the types of interior angles is always equal to 180 degrees. For a regular triangle, each interior angle volition be equal to:

180/3 = 60 degrees

lx°+sixty°+60° = 180°

Therefore, no matter if the triangle is an astute triangle or birdbrained triangle or a correct triangle, the sum of all its interior angles will e'er be 180 degrees.

Interior Angles of Quadrilaterals

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

Square
Rectangle
Parallelogram
Rhombus
Trapezium
Kite

All the shapes listed above take iv 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 equally foursquare, each interior bending will be equal to:

360/4 = 90 degrees.

90° + ninety° + xc° + 90° = 360°

Since each quadrilateral is made up 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 instance of the pentagon, it has five sides and as well it can be formed by joining iii triangles side by side. Thus, if 1 triangle has sum of angles equal to 180 degrees, therefore, the sum of angles of 3 triangles will be:

3 x 180 = 540 degrees

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

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

540°/five = 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 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 tin can be obtained in three means. Let us discuss the three dissimilar formulas in detail.

Method 1:

If "northward" 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 angle of a polygon is given, then the formula to find the interior angle is

Interior Angle of a polygon = 180° – Exterior bending 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 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

Beneath is the proof for the polygon interior angle sum theorem

Argument:

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

To prove:

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

Proof:

Interior angles example

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 north triangles = n × 180°

From the to a higher place statement, we 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 in a higher place value in (ane), we become

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

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

Take 90 equally common, and so it becomes

The sum of the interior angles = (2n – 4) × 90°

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

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

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

Outside 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 i side of the polygon and extension of the other side. The sum of an next interior angle and outside angle for any polygon is equal to 180 degrees since they form a linear pair. Likewise, the sum of exterior angles of a polygon is e'er equal to 360 degrees.

Outside angle of a polygon = 360 ÷ number of sides

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

Solved Examples

Q.ane: 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°

Outside 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 angle

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

Hence, the polygon has 10 sides.

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

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

n = 8

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

Sum = (8-2) x 180° = six x 180° = 1080°

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

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

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

Answer: Given,

Number of sides, n = x

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

Video Lesson on Angle sum and outside angle property

Exercise Questions

Discover 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?

Often Asked Questions – FAQs

What are the interior angles of a polygon?

Interior angles of a polygon are the angles that lie at the vertices, within 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, employ the given formula:
Sum = (n-2) 10 180°
Where northward is the number of sides or number of angles of polygons.

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

To notice 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 tin can exist four 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 ten sides and 10 angles.
Sum of interior angles = (10 – 2) 10 180°
= 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 angle sum of a kite will be 360°.