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Internal angle

 

 
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Internal angle
 
 
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In geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, an interior angle (or internal angle) is an angle
Angle

In geometry and trigonometry, an angle is the figure formed by two Ray sharing a common endpoint, called the vertex of the angle . The magnitude of the angle is the "amount of rotation" that separates the two rays, and can be measured by considering the length of circular arc swept out when one ray is rotated about the vertex to coincide...
 formed by two sides of a simple polygon
Simple polygon

In geometry, a simple polygon is closed polygonal chain of line segments that do not cross each other. That is, it consists of finitely many line segments, each line segment endpoint is shared by two segments, and the segments do not otherwise intersect....
 that share an endpoint, namely, the angle on the inner side of the polygon. A simple polygon has exactly one internal angle per vertex
Vertex (geometry)

In geometry, a vertex is a special kind of point which describes the corners or intersections of geometric shapes. Vertices are commonly used in computer graphics to define the corners of surfaces in 3d models, where each such point is given as a vector....
.

If every internal angle of a polygon is less than 180°, the polygon is called convex
Convex polygon

In geometry, a polygon can be either convex or concave....
.

In contrast, an exterior angle (or external angle) is an angle formed by one side of a simple polygon and a line extended from an adjacent side.

The sum of the internal angle and the external angle on the same vertex is 180°.

sum of the internal angles of a simple polygon can be determined

using the formula where the variable n represents the number of sides the polygon has.






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Externalangles
In geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, an interior angle (or internal angle) is an angle
Angle

In geometry and trigonometry, an angle is the figure formed by two Ray sharing a common endpoint, called the vertex of the angle . The magnitude of the angle is the "amount of rotation" that separates the two rays, and can be measured by considering the length of circular arc swept out when one ray is rotated about the vertex to coincide...
 formed by two sides of a simple polygon
Simple polygon

In geometry, a simple polygon is closed polygonal chain of line segments that do not cross each other. That is, it consists of finitely many line segments, each line segment endpoint is shared by two segments, and the segments do not otherwise intersect....
 that share an endpoint, namely, the angle on the inner side of the polygon. A simple polygon has exactly one internal angle per vertex
Vertex (geometry)

In geometry, a vertex is a special kind of point which describes the corners or intersections of geometric shapes. Vertices are commonly used in computer graphics to define the corners of surfaces in 3d models, where each such point is given as a vector....
.

If every internal angle of a polygon is less than 180°, the polygon is called convex
Convex polygon

In geometry, a polygon can be either convex or concave....
.

In contrast, an exterior angle (or external angle) is an angle formed by one side of a simple polygon and a line extended from an adjacent side.

The sum of the internal angle and the external angle on the same vertex is 180°.

Calculation

The sum of the internal angles of a simple polygon can be determined

using the formula where the variable n represents the number of sides the polygon has. If the polygon is also regular
Regular polygon

A regular polygon is a polygon which is Equiangular polygon and equilateral . Regular polygons may be convex or Star polygon....
 or equiangular
Equiangular polygon

File:Rectangle definition.svgIn Euclidean geometry, an equiangular polygon is a polygon whose vertex angles are equal. If the lengths of the sides are also equal then it is a regular polygon....
, dividing the result by n gives the measure of each angle.

Similarly, the measures of a given polygon's exterior angles can be calculated by dividing 360° by the number of sides of the polygon n.

External links

  • and With interactive animation
  • with interactive applets that are also useful in a classroom setting. Math Open Reference