Sum of Interior Angles of a Polygon and Formulas
A polygon is a closed geometric figure which has only two dimensions (length and width). All the vertices, sides and angles of the polygon lie on the same plane. Hence it is a plane geometric figure. At the point where any two adjacent sides of a polygon meet (vertex), the angle of separation is called the interior angle of the polygon. A polygon with three sides has 3 interior angles, a polygon with four sides has 4 interior angles and so on. Sum of all the interior angles of a polygon with ‘p’ sides is given as
Sum of Interior Angles of a Polygon Formula
The formula for finding the sum of the interior angles of a polygon is devised by the basic ideology that the sum of the interior angles of a triangle is 1800. The sum of the interior angles of a polygon is given by the product of two less than the number of sides of the polygon and the sum of the interior angles of a triangle. If a polygon has ‘p’ sides, then
Sum of interior angles = (p - 2) 1800
Sum of Interior Angles of a Regular Polygon and Irregular Polygon
A regular polygon is a polygon whose sides are of equal length. Examples of a regular polygon are equilateral triangle, square, regular pentagon etc. An irregular polygon is a polygon with sides having different lengths. Though the sum of interior angles of a regular polygon and irregular polygon with the same number of sides is the same, the measure of each interior angle differs. In the case of regular polygons, the measure of each interior angle is congruent to the other.
A Theorem about Interior Angles
Here will prove the polygon interior angle sum theorem in the following paragraphs
Here's the statement:
The sum of the interior angles of a polygon has n sides equals (2n - 4) × 900.
Here are the proofs:
In interior angles, the sum equals (2n - 4)
Here's the proof:
A polygon ABCDE has n sides. Consider any point O within the polygon. OA, OB, OC are joined together.
When a polygon has "n" sides, it forms "n" triangles.
Summarizing the angles of a triangle yields a 180-degree angle, thus, n times 1800 is the sum of the angles of n triangles.
We can conclude that based on the above statement that Total angles in the interior + sum of angles in the interior = 2-n * 900 ----(1)
Nevertheless, the angles at O sum to 360 degrees
In (1), substituting the above value gives, 360 degrees + 2n * 90 degrees = total interior angles, therefore, the interior angles sum up to (2n × 900) – 3600, assuming that 90 is common, it becomes a product of (2n - 4) * 900 equals the sum of the interior angles. In other words, the sum of inner angles of “n” is (2n - 4) × 900.
Consequently, each interior angle of a regular polygon is ((2n – 4) × 900) / n
Regular polygons have the same measures for all interior angles.
However, in the case of irregular polygons, the interior angles do not give the same measure.
The measure of each interior angle of a regular polygon is equal to the sum of interior angles of a regular polygon divided by the number of sides.
The sum of interior angles of a regular polygon and irregular polygon examples is given below.
Sum of Interior Angles of a Polygon with Different Number of Sides:
Sum of Interior Angles of a Polygon Formula Example Problems:
1. The sum of the interior angles of a regular polygon is 30600. Find the number of sides in the polygon.
Solution:
Sum of interior angles of a polygon with ‘p’ sides is given by:
Sum of interior angles = (p - 2) 1800
30600 = (p - 2) 1800
p - 2 = 30600 / 1800
p - 2 = 17
p = 17 + 2
p = 19
The polygon has 19 sides.
2. Find the value of ‘x’ in the figure shown below using the sum of interior angles of a polygon formula.
Solution:
The figure shown above has three sides and hence it is a triangle. Sum of interior angles of a three-sided polygon can be calculated using the formula as:
Sum of interior angles = (p - 2) 1800
600 + 400 + (x + 83)0 = (3 - 2) 1800
1830 + x = 1800
x = 1800 - 183
x = -3
Fun Facts:
Polygons are also classified as convex and concave polygons based on whether the interior angles are pointing inwards or outwards.
The name of the polygon generally indicates the number of sides of the polygon.
The properties of Interior angles of a Polygon are important to learn. It prepares the base of concepts for geometric figures and students can proceed to understand advanced theories later with it.
FAQs on Interior Angles of a Polygon
1. What are polygons? How are they classified?
A polygon is a closed geometric figure with several sides, angles, and vertices. A polygon is a plane geometric figure. Polygons are broadly classified into types based on the length of their sides. If a polygon has all the sides of equal length then it is called a regular polygon. Examples of regular polygons are equilateral triangles and squares. Irregular polygons are polygons with different lengths of sides. Based on the number of sides, the polygons are classified into several types. A polygon with three sides is called a triangle, a polygon with 4 sides is a quadrilateral, a polygon with five sides is a pentagon, a polygon with 6 sides is a hexagon, and soon.
2. What is the sum of interior angles of a polygon formula?
The Interior angle of a polygon is the angle formed at the point of contact of any two adjacent sides of a polygon. A polygon will have the number of interior angles equal to the number of sides it has. i.e. If a polygon has 5 sides, it will have 5 interior angles. The Sum of all the interior angles of a polygon is equal to the product of a straight angle and two less than the number of sides of the polygon. In a regular polygon, all the interior angles measure the same and hence can be obtained by dividing the sum of the interior angles by the number of sides.
Sum of interior angles = (p - 2) 1800
3. Which angles make up the interior of a polygon?
Angles formed by joining two rays at their common endpoints are called interior angles of polygons in mathematics. Inside any shape, there are interior angles. In geometry, polygons are squared shapes with sides and vertices. Each interior angle of a regular polygon is equal. For instance, all the angles in a square are equal to the right angle, or 90 degrees.
There are an equal number of interior angles for a polygon. An angle usually takes the form of a degree or a radian. When a polygon has four sides, then it will also have four angles. Moreover, different polygons have different interior angles.
4. How do you calculate interior angles?
As long as the inside angles of a polygon lie inside it, the interior angles will always lie inside of it. Three different methods can be used to obtain the formula. Let's take a look at each method in more detail.
Method 1:
Here's the formula for polygons with an arbitrary number of sides:
A Regular Polygon's interior angles are defined as "1800(n) - 3600" / n
Method 2:
To calculate the interior angle of a polygon, we take the exterior angle as an input and then apply the following formula
Observe that the interior angle of a polygon is equal to 1800 minus the exterior angle of the polygon
Method 3:
A regular polygon can be divided by its number of sides to calculate its interior angle if we know the sum of all its interior angles.
An interior angle is equal to the sum of all interior angles of one or more polygons / n
In other words "n" refers to the number of sides of the polygons.
5. A regular polygon would have how many sides of each interior angle was equal to 144 degrees?
Here's the solution:
A regular polygon would have how many sides
According to the formula,
Exterior angle plus Interior angle = 1800
Outside angle = 1800 - 1440
Hence 360 is the exterior angle
You can determine the number of sides of a regular polygon by using the following formula:
The number of sides of normal polygons is equal to 3600 / the degree of each exterior angle,
In other words, there are 10 sides since 3600 / 360 = 10 sides
Hence, the polygon has 10 sides.