What are Polygon Exterior Angles?

Exterior angles of a polygon are formed when one of its sides is extended outwards. They are the angles between the extended side and the adjacent side of the polygon. These angles are crucial for understanding the geometry of polygons and have several important properties that make them useful in various calculations and real-world applications.

Exterior Angle = 360° / n

This formula calculates the measure of a single exterior angle for a regular polygon, where 'n' is the number of sides. Since all exterior angles in a regular polygon are equal, dividing the total sum of exterior angles (360°) by the number of sides gives you the measure of each individual exterior angle.

Interior Angle = 180° - Exterior Angle

At any vertex of a polygon, the interior angle and its corresponding exterior angle form a linear pair, meaning they lie on a straight line and add up to 180°. This relationship allows you to easily find one if you know the other.

Sum of Interior Angles = (n - 2) × 180°

This formula gives the total sum of all interior angles for any polygon with 'n' sides. It's derived from the fact that any polygon can be divided into (n-2) triangles, and each triangle's angles sum to 180°.

Sum of Exterior Angles = 360°

This is a fundamental and consistent property: the sum of the exterior angles of any convex polygon (one at each vertex) is always 360°, regardless of the number of sides. Imagine walking around the perimeter of a polygon; you make a full 360° turn by the time you return to your starting point.

where:

n is the number of sides in the polygon.
Each exterior angle is supplementary to its corresponding interior angle (they add up to 180°).
The sum of exterior angles is always 360° for any convex polygon.

Properties of Polygon Angles

Each exterior angle of a regular polygon is equal: In a regular polygon, all sides are equal in length, and all interior angles are equal. Consequently, all exterior angles are also equal, making calculations straightforward.
The sum of one exterior and one interior angle at any vertex is 180°: This is because they form a linear pair on a straight line. This supplementary relationship is key to converting between interior and exterior angle measures.
The measure of each exterior angle decreases as the number of sides increases: As a polygon gains more sides, it becomes "rounder," and the turns you make at each vertex (the exterior angles) become smaller. For example, a triangle has 120° exterior angles, while a square has 90°.
As the number of sides approaches infinity, the exterior angle approaches 0°: Conceptually, a polygon with an infinite number of sides becomes a circle. At any point on a circle, the "turn" is infinitesimally small, hence the exterior angle approaches zero.
The sum of all exterior angles is constant (360°) regardless of the number of sides: This is a powerful and consistent property. Whether it's a triangle, a square, or a 100-sided polygon, if you add up all its exterior angles, the total will always be 360°.
In regular polygons, all interior angles are equal, and all exterior angles are equal: This symmetry simplifies many geometric problems and is a defining characteristic of regular polygons.

Applications and Real-world Examples

Architecture and Construction

Exterior angles are vital in designing buildings, especially for roofs, walls, and foundations. Architects and builders use these calculations to ensure structural integrity, proper drainage, and aesthetic appeal. For instance, determining the pitch of a roof or the angles for cutting materials like wood or metal for precise fits.

Navigation and Surveying

In navigation, exterior angles are important for calculating turning angles and planning routes for ships, aircraft, or even land vehicles. Surveyors use them to map land, define property boundaries, and ensure accurate measurements when laying out roads or structures.

Computer Graphics and Game Development

Polygons are the fundamental building blocks of 3D models in computer graphics and video games. Understanding exterior angles is essential for creating realistic shapes, rendering surfaces, and ensuring objects behave correctly within a virtual environment, from character models to landscapes.

Engineering and Robotics

Engineers use polygon angles in mechanical design, such as designing gears, linkages, and machine parts. In robotics, precise control of robotic arms and movements relies on calculating angles at joints, which often form polygonal chains, to ensure accurate and efficient operation.

Special Cases and Considerations

Triangle (3 sides): For a regular triangle (equilateral), each exterior angle is 360°/3 = 120°. This is the largest possible exterior angle for a regular polygon.
Square (4 sides): For a regular quadrilateral (square), each exterior angle is 360°/4 = 90°. This means the exterior angle is equal to the interior angle.
Regular Pentagon (5 sides): Each exterior angle is 360°/5 = 72°. This is a common shape found in nature and design.
Regular Hexagon (6 sides): Each exterior angle is 360°/6 = 60°. Hexagons are known for their efficiency in packing, seen in beehives and certain crystal structures.
Regular Octagon (8 sides): Each exterior angle is 360°/8 = 45°. Octagons are often used in stop signs and architectural elements.
Convex vs. Concave Polygons: While the sum of exterior angles is always 360° for convex polygons, the concept of exterior angles becomes more complex for concave polygons (those with at least one interior angle greater than 180°). Our calculator focuses on convex polygons.

Polygon Exterior Angle Calculator

Understanding Polygon Exterior Angles