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.

internal and external angles

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:

$$n-2$$

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:

$$180(n-2)$$

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:

$${\frac {180(n-2)}{n}}$$

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:

$${\frac {360}{n}}$$

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:

$$n[(n'-2){\frac {180}{n'}}]$$

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