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How to Find Exterior Angles of a Polygon? Learn how to find exterior angles of any polygon with step-by-step instructions. Master polygon geometry effortlessly.
Contents
How to Find Exterior Angles of a Polygon?
To find the exterior angles of a polygon, you can follow these steps:
Understand the Basics:
An exterior angle of a polygon is an angle formed by extending one side of the polygon beyond its vertex.
The sum of the exterior angles of any polygon is always 360 degrees. This is known as the Exterior Angle Sum Theorem.
Count the Sides:
Determine the number of sides (n) of the polygon. This information is essential for finding individual exterior angles.
Find the Exterior Angle:
To find the measure of a single exterior angle, divide 360 degrees by the number of sides (n).
- Exterior Angle = 360 degrees / n
Calculate Individual Exterior Angles:
If you want to find the measure of each exterior angle, divide 360 degrees by the number of sides (n). This will give you the measure of a single exterior angle, and all the exterior angles in the polygon will have the same measure.
For example:
If you have a regular hexagon (a polygon with 6 equal sides), you can find the exterior angle as follows:
- Exterior Angle = 360 degrees / 6 = 60 degrees
So, each exterior angle of the hexagon is 60 degrees.
If you have a non-regular polygon, you can still find the exterior angles by using the same method. The key is to know the number of sides in the polygon.
Remember that the Exterior Angle Sum Theorem always holds true for any polygon. The sum of all the exterior angles will be 360 degrees, regardless of the type of polygon you are dealing with.
What are Exterior Angles?
Exterior angles are a fundamental concept in geometry, specifically in the context of polygons. An exterior angle of a polygon is an angle formed by one side of the polygon and the extension of an adjacent side. In other words, it’s an angle that is outside the polygon.
Here are some key points about exterior angles:
Exterior Angle Theorem: The sum of an exterior angle and an interior angle at the same vertex of a polygon is always 180 degrees. In mathematical terms, if ∠A is an interior angle of a polygon at vertex A, and ∠B is the corresponding exterior angle at vertex B, then ∠A + ∠B = 180 degrees.
Exterior Angles in Different Polygons:
- In a triangle, every exterior angle is supplementary to its adjacent interior angle.
- In a quadrilateral, the sum of the exterior angles is always 360 degrees.
- In a pentagon, the sum of the exterior angles is 360 degrees as well.
- In a hexagon, the sum of the exterior angles is 360 degrees.
Exterior Angles in Regular Polygons: In a regular polygon, where all sides and angles are congruent (equal), each exterior angle has the same measure. The measure of an exterior angle in a regular polygon can be calculated using the formula: 360 degrees divided by the number of sides in the polygon.
Exterior Angles in Convex and Concave Polygons: In convex polygons, all exterior angles are directed outward from the polygon, while in concave polygons, some exterior angles point inward.
Understanding exterior angles is essential when working with polygons, and they are often used in geometry to solve various problems related to angles and shapes.
What is the Exterior Angle of a Polygon?
The exterior angle of a polygon is an angle formed by one side of the polygon and the extension of an adjacent side. In other words, if you have a polygon, such as a triangle, quadrilateral, pentagon, etc., and you extend one of its sides, the angle formed where the extension intersects the adjacent side is called the exterior angle.
The sum of the exterior angles of any polygon is always 360 degrees. This is known as the Exterior Angle Theorem. The Exterior Angle Theorem is a fundamental concept in geometry and is used in various geometric proofs and calculations. It’s particularly useful for solving problems related to angles and sides in polygons.
Exterior Angles of Polygon Theorem
The Exterior Angles of a Polygon Theorem, also known as the Exterior Angle Theorem, is a fundamental principle in geometry that relates the exterior angles of a polygon to its interior angles. This theorem states that:
The measure of an exterior angle of a polygon is equal to the sum of the measures of its two remote interior angles.
Here are some key terms in this theorem:
Exterior Angle: An exterior angle of a polygon is formed by extending one side of the polygon beyond the vertex. It is the angle formed between the extension of one side and the adjacent side.
Interior Angle: An interior angle of a polygon is an angle formed by two adjacent sides of the polygon inside the figure.
Remote Interior Angles: The two interior angles that are not adjacent to the exterior angle are called remote interior angles.
Mathematically, if “x” represents the measure of an exterior angle, and “a” and “b” represent the measures of the two remote interior angles, then the Exterior Angle Theorem can be expressed as:
This theorem is particularly useful when you need to find the measure of an exterior angle of a polygon or when working with various geometric properties of polygons. It is applicable to all types of polygons, whether they are regular or irregular.
In summary, the Exterior Angle Theorem provides a relationship between the measures of exterior angles and the corresponding interior angles in a polygon, helping in various geometric calculations and proofs.
How Many Degrees is the Exterior of a Polygon?
The sum of the exterior angles of any polygon, regardless of the number of sides it has, is always 360 degrees. This means that if you were to go around the exterior of a polygon and measure the angles formed at each vertex, and then add up all those angle measurements, the total would be 360 degrees. This is a fundamental property of polygons, and it holds true for all convex and concave polygons.
Solved Examples on Exterior Angles of a Polygon
Exterior angles of a polygon are the angles formed outside the polygon at each vertex. These angles can be found using the properties of polygons. Here are some solved examples:
Example 1: Find the sum of the exterior angles of a pentagon.
A pentagon has 5 sides, and each exterior angle corresponds to one of its vertices. The sum of the exterior angles of any polygon is always 360 degrees. So, in this case, the sum of the exterior angles is 360 degrees.
Example 2: Find the measure of each exterior angle of a regular hexagon.
A regular hexagon has 6 sides, and all its exterior angles are equal because all its interior angles are also equal. To find the measure of each exterior angle, you can divide the total sum of exterior angles (360 degrees) by the number of exterior angles (6):
Each exterior angle of a regular hexagon measures 60 degrees.
Example 3: Find the measure of each exterior angle of a triangle.
A triangle has 3 sides. To find the measure of each exterior angle, you can use the formula for the sum of exterior angles in any polygon, which is 360 degrees, and then divide it by the number of exterior angles:
Each exterior angle of a triangle measures 360 degrees / 3 = 120 degrees.
Example 4: Find the measure of each exterior angle of a quadrilateral.
A quadrilateral has 4 sides. To find the measure of each exterior angle, use the same formula for the sum of exterior angles:
Each exterior angle of a quadrilateral measures 360 degrees / 4 = 90 degrees.
Example 5: Find the measure of each exterior angle of an irregular pentagon with exterior angles measuring 45 degrees, 60 degrees, 75 degrees, 90 degrees, and 105 degrees.
In an irregular polygon, the exterior angles can have different measures. To find the measure of each exterior angle, you can use the formula:
Sum of exterior angles = 360 degrees.
So, the sum of the exterior angles is 45 + 60 + 75 + 90 + 105 = 375 degrees.
Now, divide this sum by the number of exterior angles (5):
Each exterior angle of the irregular pentagon measures 375 degrees / 5 = 75 degrees.
These examples demonstrate how to find the sum of exterior angles in different polygons and how to calculate the measure of each exterior angle.
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