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Internal and external angles
Internal and external angles
Internal and external angles
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"Interior angle" redirects here. For interior angles on the same side of the transversal, see Transversal line.
The corresponding internal (teal) and external (magenta) angles of a polygon are supplementary (sum to a half turn). The external angles of a non-self-intersecting closed polygon always sum to a full turn.
Types of angles
2D angles

Right
Interior
Exterior

Spherical
2D angle pairs

Adjacent
Vertical
Complementary
Supplementary

Transversal
3D angles

Dihedral

Solid
Internal and external angles

In geometry, an angle of a polygon is formed by two adjacent sides. For a simple polygon (non-self-intersecting), regardless of whether it is convex or non-convex, this angle is called an internal angle (or interior angle) if a point within the angle is in the interior of the polygon. A polygon has exactly one internal angle per vertex.

If every internal angle of a simple polygon is less than a straight angle (π radians or 180°), then the polygon is called convex.

In contrast, an external angle (also called a turning angle or exterior angle) is an angle formed by one side of a simple polygon and a line extended from an adjacent side.[1]: pp. 261–264 

Properties

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  • The sum of the internal angle and the external angle on the same vertex is π radians (180°).
  • The sum of all the internal angles of a simple polygon is π(n − 2) radians or 180(n − 2) degrees, where n is the number of sides. The formula can be proved by using mathematical induction: starting with a triangle, for which the angle sum is 180°, then replacing one side with two sides connected at another vertex, and so on.
  • The sum of the external angles of any simple polygon, if only one of the two external angles is assumed at each vertex, is 2π radians (360°).
  • The measure of the exterior angle at a vertex is unaffected by which side is extended: the two exterior angles that can be formed at a vertex by extending alternately one side or the other are vertical angles and thus are equal.

Extension to crossed polygons

[edit]

The interior angle concept can be extended in a consistent way to crossed polygons such as star polygons by using the concept of directed angles. In general, the interior angle sum in degrees of any closed polygon, including crossed (self-intersecting) ones, is then given by 180(n − 2k)°, where n is the number of vertices, and the strictly positive integer k is the number of total (360°) revolutions one undergoes by walking around the perimeter of the polygon. In other words, the sum of all the exterior angles is 2πk radians or 360k degrees. Example: for ordinary convex polygons and concave polygons, k = 1, since the exterior angle sum is 360°, and one undergoes only one full revolution by walking around the perimeter.

Extension to polyhedra

[edit]
Further information: Descartes' theorem on total angular defect

Consider a polyhedron that is topologically equivalent to a sphere, such as any convex polyhedron. Any vertex of the polyhedron will have several facets that meet at that vertex. Each of these facets will have an interior angle at that vertex and the sum of the interior angles at a vertex can be said to be the interior angle associated with that vertex of the polyhedron. The value of 2π radians (or 360 degrees) minus that interior angle can be said to be the exterior angle associated with that vertex, also known by other names such as angular defect. The sum of these exterior angles across all vertices of the polyhedron will necessarily be 4π radians (or 720 degrees), and the sum of the interior angles will necessarily be 2π(n − 2) radians (or 360(n − 2) degrees) where n is the number of vertices. A proof of this can be obtained by using the formulas for the sum of interior angles of each facet together with the fact that the Euler characteristic of a sphere is 2.

For example, a rectangular solid will have three rectangular facets meeting at any vertex, with each of these facets having a 90° internal angle at that vertex, so each vertex of the rectangular solid is associated with an interior angle of 3 × 90° = 270° and an exterior angle of 360° − 270° = 90°. The sum of these exterior angles over all eight vertices is 8 × 90° = 720°. The sum of these interior angles over all eight vertices is 8 × 270° = 2160°.

References

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Internal and external angles

View on Grokipedia
from Grokipedia
In , internal angles (also known as interior angles) are the angles formed at the vertices inside a , while external angles (or exterior angles) are the angles formed outside the by extending one of its sides beyond a vertex and measuring the between that extension and the adjacent side. For a with nn sides, the sum of the interior angles is (n−2)×180∘(n-2) \times 180^\circ, a formula derived from triangulating the into n−2n-2 triangles, each contributing 180∘180^\circ. In contrast, the sum of the exterior angles of any is always 360∘360^\circ, regardless of the number of sides, as these angles collectively form a full around a point when traversed along the 's perimeter. These measures are fundamental in classifying as regular or irregular and in applications such as , , and , where precise angular calculations ensure structural integrity or accurate rendering. For regular , where all sides and are equal, each interior measures (n−2)×180∘n\frac{(n-2) \times 180^\circ}{n} and each exterior measures 360∘n\frac{360^\circ}{n}, facilitating symmetry-based designs. Understanding the relationship between interior and exterior —at each vertex, they form a straight line summing to 180∘180^\circ—enables derivations of properties and proofs of geometric theorems.

Basic Concepts

Internal angles

A is a closed plane figure bounded by a finite number of straight line segments. The internal angle of a is the angle formed by two adjacent sides at a vertex, lying inside the boundary of the . These angles define the shape's turning at each corner and are fundamental to understanding polygonal geometry. Internal angles are measured in units such as degrees or radians. In degrees, common examples include the equilateral triangle, where each internal angle measures 60 degrees; the square, with each at 90 degrees; and the regular pentagon, featuring 108 degrees per internal angle. Visually, internal angles in convex polygons appear at the vertices as the inward-facing angles between consecutive sides, enclosing the polygon's interior region. For instance, in a drawn triangle or quadrilateral, these angles are the ones oriented toward the center of the shape, distinguishing them from outward extensions. External angles serve as their supplementary counterparts at each vertex.

External angles

An external angle of a polygon is the angle formed by one side of the polygon and the extension of an adjacent side, measured outward from the vertex. This angle represents the turning angle required to follow the polygon's boundary when traversing its perimeter. To construct an external angle, extend one side of the polygon beyond a vertex, then measure the angle between this extension and the adjacent side emanating from that vertex. The measurement is taken in the outward direction, away from the polygon's interior, and for consistency across all vertices, external angles are always oriented in the same rotational sense—either all clockwise or all counterclockwise—ensuring a uniform traversal around the shape. This directionality distinguishes the external angle from the internal angle at the same vertex, which points inward. For example, in an , where each internal angle measures 60°, the external angle at any vertex is 120°. Similarly, in a square, with internal angles of 90°, each external angle is 90°. These examples illustrate how external angles capture the supplementary relationship to internal angles at each vertex, though the focus here is on their independent outward formation. Diagrams of external angles typically show the polygon with one side extended as a dashed line, highlighting the acute or obtuse turn outside the shape, in contrast to the internal angle's inward bend within the boundary.

Properties in Simple

Supplementary relationship

In any , the internal angle and the corresponding external angle at a given vertex are supplementary, meaning their measures add up to 180° (or π radians), as they are adjacent angles formed on a straight line. This core property stems from the geometric construction of the external angle, where one side of the is extended beyond the vertex. The derivation relies on the straight-line postulate in Euclidean geometry, which states that the sum of adjacent angles forming a linear pair equals 180°. Specifically, extending a side creates a straight line at the vertex, positioning the internal angle and external angle as adjacent angles along this line; thus, their measures must sum to 180°. A simple proof involves drawing the extension and identifying the linear pair: the internal angle θ and external angle φ satisfy θ + φ = 180° by definition of adjacent angles on a straight line. This relationship is expressed mathematically as: θ+ϕ=180∘\theta + \phi = 180^\circ where θ denotes the internal angle and φ the external angle at the vertex. The supplementary relationship holds precisely for convex polygons, where all internal angles are less than 180°. In concave polygons featuring reflex internal angles greater than 180°, the external angle adjusts to maintain the sum of 180°, often resulting in a negative measure to account for the directional turn. For instance, in , each internal measures 90°, so the corresponding external is 90°, yielding 90° + 90° = 180°. In an irregular with an internal of 120° at one vertex, the external there measures 60°, again summing to 180°.

Angle sums

The sum of the interior angles of a simple with nn sides, where n≥3n \geq 3, is (n−2)×180∘(n-2) \times 180^\circ. This formula arises from triangulating the , which divides it into n−2n-2 s; since each has an interior angle sum of 180∘180^\circ, the total is (n−2)×180∘(n-2) \times 180^\circ. The underlying principle for the triangular case traces to , specifically Proposition I.32 in Euclid's Elements, which establishes that the interior angles of any sum to 180∘180^\circ (or two right angles). For the exterior angles of a simple —defined as the angles formed by extending one side at each vertex—the sum is always 360∘360^\circ (or 2π2\pi radians), regardless of nn or whether the is convex or concave. This fixed total reflects the net turning when traversing the 's boundary, which completes exactly one full of 360∘360^\circ. The consistency arises because each exterior measures the deviation from a straight line (180∘180^\circ), and their directed sum accounts for the overall closure of the path. Examples illustrate these sums clearly. A triangle (n=3n=3) has an interior angle sum of (3−2)×180∘=180∘(3-2) \times 180^\circ = 180^\circ and an exterior sum of 360∘360^\circ. A quadrilateral (n=4n=4) has an interior sum of 360∘360^\circ and the same exterior sum of 360∘360^\circ. In regular polygons, where all sides and angles are equal, each exterior angle is 360∘/n360^\circ / n, ensuring the total remains 360∘360^\circ. This uniformity in the exterior sum holds due to the supplementary relationship between interior and exterior angles at each vertex, leading to the global balance.

Extensions and Generalizations

Crossed polygons

Crossed polygons, also known as self-intersecting polygons, are non-simple polygonal figures in which one or more sides cross over each other, forming intersections that are not vertices of the polygon, such as the or other star polygons. Unlike simple polygons, which do not intersect themselves, crossed polygons require adjustments to traditional sum formulas to account for the topology introduced by these crossings. The sum of the interior angles in a crossed is given by the formula S=180∘(n−2k)S = 180^\circ (n - 2k), where nn is the number of vertices and kk is the , representing the number of full 360° rotations made while traversing the perimeter. This kk equals 1 for simple (convex or concave) polygons but increases to 2 or more for crossed figures, effectively reducing the interior angle sum compared to the simple case. For instance, in a star pentagon denoted by the {5/2}, n=5n = 5 and k=2k = 2, yielding an interior angle sum of 180∘(5−4)=180∘180^\circ (5 - 4) = 180^\circ. External angles in crossed polygons are typically considered as directed turning angles at the vertices. The sum of these directed external angles equals 360∘k360^\circ k, where kk is the same (also known as the in the context of regular star polygons). In the example with density 2, the external angles sum to 720∘720^\circ, reflecting the additional rotation due to the self-intersections. This contrasts with simple polygons, where the external angle sum is always 360∘360^\circ regardless of nn. To illustrate, consider a simple , which has n=5n = 5 and k=1k = 1, resulting in an interior angle sum of 540∘540^\circ and external sum of 360∘360^\circ. In contrast, the star {5/2} with k=2k = 2 has the adjusted sums of 180∘180^\circ for interiors and 720∘720^\circ for exteriors, as verified through geometric constructions where vertices can be varied while preserving the angle totals. Diagrams of such star polygons typically highlight the interior angles at the five vertices separately from the supplementary angles formed at intersection points along the sides, emphasizing that only vertex angles contribute to the interior sum.

Polyhedra

In three-dimensional polyhedra, internal angles at a vertex are defined as the planar angles formed by the edges within the polygonal faces meeting at that vertex. These face angles contribute to the local , with the external angle at each such planar junction being the complement to 180° within the face's local plane. For a convex polyhedron, the sum of the internal face angles at any vertex must be less than 360° to ensure the faces enclose space without gaps or overlaps. The global sum of all internal face angles across a polyhedron homeomorphic to a sphere is given by 360∘×(V−2)360^\circ \times (V - 2), where VV is the number of vertices; this follows from the sum of internal angles in each face, combined with Euler's formula V−E+F=2V - E + F = 2 (with EE edges and FF faces). Consequently, the sum of the angular defects—defined as 360∘360^\circ minus the internal angle sum at each vertex, analogous to external turning angles—totals 720∘720^\circ for such polyhedra. For example, in a , three square faces meet at each of the 8 vertices, with each face contributing a 90° internal , yielding a per-vertex sum of 270° and a defect of 90°; the total internal sum is thus 360∘×6=2160∘360^\circ \times 6 = 2160^\circ, and the defects sum to 720∘720^\circ. In a regular , three equilateral triangular faces meet at each of the 4 vertices, each contributing 60°, for a per-vertex sum of 180° and defect of 180°; the total internal sum is 360∘×2=720∘360^\circ \times 2 = 720^\circ, with defects again totaling 720∘720^\circ. These relations hold for convex polyhedra and star polyhedra, both of which possess spherical topology ( 2). For polyhedra with hyperbolic topology ( less than 2), such as those on higher-genus surfaces, the total angular defect scales with the as 720∘×χ/2720^\circ \times \chi / 2, resulting in smaller or negative totals that permit different local angle configurations.

References

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