Students can know more about polygon within its sides or we can say n-sided polygon in Rectilinear Figures. An interior angle is an angle inside the polygon at one of its vertices. Get to know the sum of the measures of the interior angles of a polygon with n sides.
You can see the Sum of the Interior Angles of an n-sided Polygon has a theorem with step by step explanation from this page. Along with the theorem, you can find interior angles of a n sided polygon formula, examples of the sum of the interior angles here.
Sum of the Interior Angles of an n-sided Polygon Theorem & Proof
Theorem:
Prove that the sum of the interior angles of a polygon of n sides is equal to (2n – 4) right angles?
Proof:
Given
Let ABCD………Z be a polygon of n sides
To Prove
∠A + ∠B + ∠C + ∠D +………+ ∠Z = (2n – 4)90°
Take any point O inside the polygon join OA, OB, OC, OD…… OZ.
Proof
The polygon has n sides, n triangles are formed they are ∆OAB, ∆BC,….. ∆OZA [ therefore on each side of the polygon one triangle has been drowned]
The sum of all the angles of the n triangles is 2n right angles [ the sum of the angles of each triangle is 2 right angles]
∠A + ∠B + ∠C + ∠D +……..+ ∠Z( sum of all angles formed at O = 2n right angles) [ from above statement]
∠A + ∠B +∠C + ∠D +………+∠Z + 4 right angles= 2 right angles [ sum of angles around the point O is 4 right angles]
∠A + ∠B + ∠C + ∠D +……..+∠Z = 2n right angles – 4 right angles [ from the above equation]
(2n – 4) right angles
(2n – 4)90°
Hence proved
Sum of the Interior Angles of an n-sided Polygon Examples
Example 1.
Find the sum of the interior angles of 6 sides.
Solution:
Here n = 6
Sum of the interior angles = (2n – 4) × 90°
=(2(6)-4) × 90°
12 – 4 × 90°
8 × 90°
720°
Therefore the sum of the interior angles of a polygon is 720°
Example 2.
Sum of the interior angles of the polygon is 180°. Find the number of sides of the polygon.
Solution:
Let the number of sides = n
Therefore (2n – 4) × 90° = 180°
2n – 4 = 180/90
2n – 4 = 2
2n = 2 + 4
2n = 6
n = 3
Therefore the number of sides of the polygon is 3.
Example 3.
The ratio of the number of sides of two regular polygons is 1:4 and the ratio of the sum of their interior angles is 4:8. Find the number of sides of each polygon.
Solution:
Let the number of sides of the two regular polygon be n1 and n2
According to the problem
n1/n2 = 1/4
n1 = n2/4
Again
2(n1 – 2) × 90°/2(n2 – 2) × 90° = 2/4
4(n1 – 2) = 2(n2 – 4)
4n1 = 2n2 + 4
4(n2/4) = 2n2 + 4
n2 = 2n2 + 4
n2 = 4
n1 = n2/4 = 4/4 = 1
Therefore the number of sides of the regular polygon is 4 and 1.
Example 4.
Find the sum of the interior angles of the 4 sides.
Solution:
Here n = 4
Sum of the interior angles = (2n – 4) × 90°
=(2(4)-4) × 90°
8 – 4 × 90°
4 × 90°
360°
Therefore the sum of the interior angles of a polygon is 360°
Example 5.
Sum of the interior angles of the polygon is 720°. Find the number of sides of the polygon.
Solution:
Given,
The sum of the interior angles of the polygon is 720°.
Let the number of sides = n
Therefore (2n – 4) × 90° = 720°
2n – 4 = 720/90
2n – 4 = 8
2n = 8 + 4
2n = 12
n = 6
Therefore the number of sides of the polygon is 6.
FAQs on Sum of the Interior Angles of an n-sided Polygon
1. What is the sum of all interior angles of a regular polygon of n sides?
The sum of all interior angles of a regular polygon of n sides is 180(n – 2).
2. What is the formula for an N-sided polygon?
S = (n – 2) × 180°
Where,
S = sum of the interior angles
3. What is the sum of the interior angle of a pentagon?
The sum of the interior angle of a pentagon is 540°