# Supplementary mathematics/Internal and external angle

An interior angle or interior angle of a polygon is another type of angle used to measure the interior angle of a regular polygon.

An exterior angle is another type of angle used to measure the exterior angles of regular polygons.

## Features

• In an interior angle, the greater the number of sides, the greater the size of the interior angle.
• In the exterior angle, the more the number of sides, the smaller the exterior angle.
• If we add the internal angle to the external angle, it becomes equal to 180 degrees.
• Internal angle and external angle are complementary to each other.
• The sum of interior angles depends on the number of sides of a regular polygon
• The sum of the external angles of a regular polygon is always equal to 360 degrees

An equilateral triangle is the only regular polygon whose exterior angle is greater than its interior angle.

• The sum of the interior angles of a triangle is 180 degrees.

A square is the only regular polygon whose interior angle is equal to its exterior angle.

## internal angle measurement

Consider a regular n-gon.

First, we calculate the number of triangles according to the famous polygons

• Square: 2 triangles
• Regular pentagon: 3 triangles
• Regular hexagon: 4 triangles
• Regular heptagon: 5 triangles
• Regular octagon: 6 triangles
• Regular octagon: 7 triangles
• Regular decagon: 8 triangles

According to this model, we find that the number of triangles is less than the number of sides of regular polygons. Therefore, the number of triangles inside any regular polygon is equal to this relationship.

Number of triangles: ${\displaystyle n-2}$

Because the set of interior angles of a triangle is 180 degrees, the set of interior angles is based on the sum of the angles of the number of triangles.

Sum of interior angles:${\displaystyle 180(n-2)}$

The interior angle measure of a regular polygon is equal to the number of divided sides. Because the number of vertices is equal to the number of sides.

Sum of Size of interior angles: ${\displaystyle {\frac {180(n-2)}{n}}}$

## external angle measurement

The sum of the external angles of any regular polygon is equal to 360 degrees. Therefore, to measure the exterior angle, we must divide 360 degrees by the number of sides of a regular polygon to determine the size of the angle.

Sum of external angles: 360 degrees: ${\displaystyle {\frac {360}{n}}}$

## Sum of internal angle measures of regular polyhedral faces

The sum of the internal angles of a polygon is calculated by the formula of the internal angles of the polygon itself, since its faces are regular polygons. We call this angle the interior angle of a polyhedron, which is a form of this relation.

Interior angle measure of a regular polyhedron:${\displaystyle n[(n'-2){\frac {180}{n'}}]}$

blockquote where n is equal to the number of faces and n is the number of sides of a regular polyhedron.

## TABLE OF INTERIOR ANGLES

The name of the polygon Sum of interior angles The size of the interior angle External angle size
Equilateral triangle ${\displaystyle 180}$ ${\displaystyle 60}$ ${\displaystyle 120}$
Square ${\displaystyle 360}$ ${\displaystyle 90}$ ${\displaystyle 90}$
Regular pentagon ${\displaystyle 540}$ ${\displaystyle 108}$ ${\displaystyle 72}$
Regular hexagon ${\displaystyle 720}$ ${\displaystyle 120}$ ${\displaystyle 60}$
Regular octagon ${\displaystyle 1080}$ ${\displaystyle 135}$ ${\displaystyle 45}$
Regular Nonagon ${\displaystyle 1260}$ ${\displaystyle 140}$ ${\displaystyle 40}$
Regular decagon ${\displaystyle 1440}$ ${\displaystyle 144}$ ${\displaystyle 36}$
Regular dodecahedron ${\displaystyle 1800}$ ${\displaystyle 150}$ ${\displaystyle 30}$
Regular pentagon ${\displaystyle 2340}$ ${\displaystyle 156}$ ${\displaystyle 24}$
Regular hexagon ${\displaystyle 2520}$ ${\displaystyle 157.5}$ ${\displaystyle 22.5}$
regular dodecahedron ${\displaystyle 3240}$ ${\displaystyle 162}$ ${\displaystyle 18}$
Twenty-four regular sides ${\displaystyle 3960}$ ${\displaystyle 165}$ ${\displaystyle 15}$
Regular triangle ${\displaystyle 5040}$ ${\displaystyle 168}$ ${\displaystyle 12}$
Thirty regular dodecahedrons ${\displaystyle 5400}$ ${\displaystyle 168.75}$ ${\displaystyle 11.25}$
Thirty regular hexagons ${\displaystyle 6120}$ ${\displaystyle 170}$ ${\displaystyle 10}$
Regular quadrilateral ${\displaystyle 6840}$ ${\displaystyle 171}$ ${\displaystyle 9}$
Regular hexagon ${\displaystyle 10440}$ ${\displaystyle 174}$ ${\displaystyle 6}$
Regular octagon ${\displaystyle 15840}$ ${\displaystyle 176}$ ${\displaystyle 4}$
Regular centagon ${\displaystyle 17640}$ ${\displaystyle 176.4}$ ${\displaystyle 3.6}$
One hundred and twenty regular polygons ${\displaystyle 21240}$ ${\displaystyle 177}$ ${\displaystyle 3}$