About Interior of Angle of a Polygon
Interior Angle of a Polygon
The interior angle of a polygon lies inside the polygon and is formed by any of its two adjacent sides meeting at the vertex. The number of interior angles are equal to the number of sides of the polygon. For example, a triangle is a polygon with three sides, so the number of interior angles is three. Similarly if a polygon has 5 sides then it has five interior angles.
Sum of the Interior Angles of a Polygon
The sum of interior angle measures of a polygon is given by the formula:
Sum of interior angles, S = (n – 2) x 180°
Where n is the number of sides of the polygon.
The table below shows us the list of the sum of interior angles of different polygons.
Measure of each Interior Angle of Regular Polygons
In a regular polygon the measure of each interior angle is the same. In other words, all the interior angles of a regular polygon are equal.
So we can find the measure of each interior angle by dividing the sum of the interior angles by the number of sides, that is,
Measure of each interior angle of regular polygon = \(\frac{sum~of~interior~angles}{number~of~sides}\)
Measure of each interior angle of regular polygon = \(\frac{(n-2)~\times~180^\circ}{n}\)
Let us find the measure of interior angle of a regular quadrilateral:
The sum of the interior angles of a quadrilateral is 360° and the number of sides of a quadrilateral is 4.
Substitute these values in the above formula to get each interior angle measure:
Measure of each interior angle = \(\frac{360^\circ}{4}\) = 90°
So, each interior angle of regular quadrilateral measures 90°.
With the use of the above formula we can find the interior angle measure of any regular polygon.
The table below shows the interior angle measures of some regular polygons:
Solved Examples
Example 1: What is the measure of each interior angle of a regular octagon?
Solution:
Since an octagon has eight sides so, n = 8
Interior angle = \(\frac{(n~-~2)~\times~180^\circ}{n}\) Write the formula
Interior angle = \(\frac{(8~-~2)~\times~180^\circ}{8}\) Substitute 8 for n
Interior angle = 135° Solve
So, the measure of each interior angle of regular octagon is 135°.
Example 2: A ‘STOP’ sign board is in the shape of a regular hexagon. Find the measure of each interior angle of the sign board.
Solution:
Since hexagon has six sides so, n = 6
Interior angle = \(\frac{(n~-~2)~\times~180^\circ}{n}\) Write the formula
Interior angle = \(\frac{(6~-~2)~\times~180^\circ}{6}\) Substitute 6 for n
Interior angle = 120° Solve
So, the measure of each interior angle of the board is 120°.
Example 3: Find the sum of the interior angles of a polygon with five sides.
Solution:
Given n = 5
S = (n – 2) x 180° Formula for the sum of interior angles of polygon
S = (5 – 2) x 180° Substitute 5 for n
S = 540°
So, the sum of the interior angles of a polygon with five sides is 540°
A heptagon has seven sides, and thus, it has seven interior angles.
The measure of each interior angle of a regular triangle(equilateral triangle) is 60°.
The sum of the interior angles of a polygon is given by the formula,
S = (n – 2) times 180 degrees.
The sum of interior angles of a triangle is 180°.
A rhombus is a quadrilateral so the sum of the interior angles of a rhombus is 360 degrees.