Exterior Angles of a Polygon
An exterior angle of a polygon is the angle that is formed between any side of the polygon and a line extended from the next side. Every polygon has interior and exterior angles. The exterior is the term opposite to the interior which means outside. Therefore exterior angles can be found outside the polygon. The sum of the exterior angles of any polygon is equal to 360°. Any flat shape or figure is said to have interior or exterior angles only if it is a closed shape. For example, a triangle is a polygon, which satisfies this condition.
|1.||Exterior Angles of a Polygon Definition|
|2.||How to Find the Exterior Angles of a Polygon?|
|3.||Exterior Angles of Polygon Theorem|
|4.||FAQs on Exterior Angles of a Polygon|
Exterior Angles of a Polygon Definition
An exterior angle is defined as the angle that is formed outside the polygon between a side of the polygon and its adjacent extended side. A polygon is a flat shape or figure that is made up of at least three straight sides and three angles. The exterior angle is created between the extended line and one adjacent side of the polygon. We can also observe that the total number of sides of the polygon is equal to the total number of exterior angles in the polygon. For example, a pentagon has 5 sides, therefore, there are 5 exterior angles. Also, the sum of the exterior and the adjacent interior angles is equal to 180° since they are in a straight line. For a square, each interior angle = 90° and each exterior angle = 90°. For an equilateral triangle, each interior angle = 60° and each exterior angle = 120°. They form a linear pair.
How to Find Exterior Angles of a Polygon?
The exterior angles of a polygon sum up to 360° because the exterior angles add to one revolution of a circle. Both regular and irregular polygons have exterior angles. A regular polygon is one in which all the side lengths and angles are of equal measures, whereas an irregular polygon has sides and angles of different measures. Let us understand the case of finding exterior angles for a regular polygon and irregular polygon through examples.
Exterior Angles of a Regular Polygon
A regular hexagon has 6 sides and 6 angles. Let us assume that each exterior angle of the hexagon is equal to 'k'. Let us imagine a circle that is starting at a point(labeled as a starting point) and is going to reach the same place again. It has to travel along the boundary or the outline of the hexagon to reach again to the starting point. In this path, the circle has to make 6 turns to reach the starting point again. The circle rotates through each of the vertices of the hexagon and reaches the starting point. It means that the circle has taken one full turn, which is equal to 360°. Here the hexagon's exterior angles sum up to 360° ⇒ 6 k = 360°. Thus each exterior angle k measures 60°each.
Thus the sum of the exterior angles of any polygon is equal to 360°.
Exterior Angles of an Irregular Polygon
An irregular polygon has sides and angles of different measures. Let us take a look at the figure in which a quadrilateral has 4 unequal sides and angles. As in the case with a regular polygon, an irregular polygon also has exterior angles that add up to 360°. Each exterior angle in an irregular polygon also is (180° - its linear pair). Thus in the given figure below, we observe that the exterior angles sum up to 360°.
Let us find the formula for exterior angles of a polygon. For both regular and irregular polygons, it is to be noted that the sum of each interior and the exterior angle on a side of the polygon equals 180° since they form a linear pair. Thus we can generalize in both the cases of a regular and an irregular polygon that the exterior angle of a polygon is given by 360°/n, where 'n' is the number of sides of the polygon.
An exterior angle of a polygon = 360°/ Number of sides of the polygon.
Exterior Angles of Polygon Theorem
Theorem statement: If a polygon is a convex polygon, then the sum of its exterior angles considering one at each vertex is equal to 360°.
Proof: Let us consider a polygon with n number of sides or n-gon, where the sum of its exterior angles is N. For any closed shape that is formed by a side and vertex, the sum of the exterior angles is always equal to the sum of the linear pairs and the sum of the interior angles.
N = 180n - 180(n - 2)
N = 180n - 180n + 360
N = 360.
Therefore, the sum of the exterior angles of n vertex is equal to 360°.
Topics Related to Exterior Angles of a Polygon
Exterior Angles of a Polygon
Example 1: Find the measure of the exterior angles of an 8 sided regular polygon.
A regular polygon is one in which all the sides and angles are equal. The polygon with 8 equal sides is an octagon. The formula to find out the individual external angles of a polygon is given by,
An exterior angle of a polygon = 360°/ Number of sides of the polygon
The number of sides of an octagon is 8.
Therefore, the exterior angles of the polygon = 360°/ 8.
Therefore, each exterior angle of the octagon = 45°
Example 2: Find the measure of the missing exterior angle in the given polygon.
From the property of the sum of the exterior angles of a polygon, we know that the sum of all the exterior angles equals 360°.
Therefore, y + 50° + 70° + 112° = 360°.
y + 232° = 360°.
y = 360° - 232°.
Therefore, the missing exterior angle y = 128°.
FAQs on Exterior Angles of a Polygon
What are Exterior Angles?
Exterior angles are angles that are formed outside a polygon. An angle is formed between two lines or line segments. In a polygon, there are at least three sides and angles. An exterior angle is formed between one side of a polygon and the line that is formed by extending the side that is adjacent to it. The total number of exterior angles in a polygon is equal to the total number of sides in it. For example, a square is a polygon that has 4 exterior angles.
How to Find Exterior Angles in a Polygon?
Exterior angles in a polygon are found by using the formula 360°/Number of sides of the polygon. If there are 9 sides in the polygon, then each exterior angle in the polygon is equal to 360°/9, which is 40°. The same formula is applicable to a regular polygon and an irregular polygon.
What is the Sum of the Exterior Angles of a Polygon?
In any polygon, the sum of the exterior angles is equal to 360°. Imagine that an object is placed at one of the vertices of a polygon. At every vertex, the object takes a turn of a certain angle such that it completes one rotation. When the angles of rotation are summed up it results in 360°. This is the reason for having the sum of exterior angles of a polygon as 360°.
What are the Sum of Interior and Exterior Angles of a Polygon?
A polygon is a figure with a minimum of three straight sides and angles. The sum of an interior angle and the exterior angle of one side of a polygon is equal to 180° since both the angles lie in a straight line. If there are 3 sides in a polygon, then on each side the sum of the interior and exterior angles are equal to 180°, which means in all, the sum of the interior and exterior angles for a polygon is 180° × 3, which is equal to 540°.
Is the Sum of Exterior Angles of a Regular and an Irregular Polygon Equal to 360°?
Be it a regular or an irregular polygon, the sum of the exterior angles of the polygon are always equal to 360°.
Why is the Sum of Exterior Angles of a Polygon Always 360°?
The sum of the exterior angles of a polygon is always considered as 360° since the exterior angles are supplementary to the interior angles and they measure 130°, 110°, 120°, respectively.
How Do You Do Exterior Angle Theorem?
According to the theorem, Exterior Angle = Sum of two Opposite non-Adjacent Interior Angles.