HomeTren&dThe Sum of Exterior Angles of a Polygon: Explained and Explored

The Sum of Exterior Angles of a Polygon: Explained and Explored

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When it comes to polygons, there are many fascinating properties and characteristics to explore. One such property is the sum of exterior angles of a polygon. In this article, we will delve into the concept of exterior angles, understand how they relate to polygons, and explore the intriguing relationship between the number of sides and the sum of exterior angles. So, let’s dive in!

Understanding Exterior Angles

Before we delve into the sum of exterior angles, let’s first understand what exterior angles are. An exterior angle of a polygon is formed by extending one of its sides. In other words, it is the angle formed between a side of a polygon and the line extended from the adjacent side.

For example, consider a triangle. If we extend one of its sides, we form an exterior angle. Similarly, in a quadrilateral, pentagon, or any other polygon, extending a side will create an exterior angle.

The Relationship Between the Number of Sides and the Sum of Exterior Angles

Now that we have a basic understanding of exterior angles, let’s explore the relationship between the number of sides in a polygon and the sum of its exterior angles. This relationship is governed by a simple formula:

Sum of Exterior Angles = 360°

This formula holds true for any polygon, regardless of the number of sides it has. The sum of the exterior angles of any polygon will always be equal to 360 degrees.

Example 1: Triangle

Let’s start with a triangle, the simplest polygon. A triangle has three sides, and if we extend each side to form an exterior angle, we can measure the angles and calculate their sum.

Consider a triangle with angles A, B, and C. If we extend side AB, we form an exterior angle at C. Similarly, extending side BC creates an exterior angle at A, and extending side CA creates an exterior angle at B.

Let’s measure these exterior angles:

  • Exterior Angle at A = 180° – A
  • Exterior Angle at B = 180° – B
  • Exterior Angle at C = 180° – C

Now, let’s calculate their sum:

Sum of Exterior Angles = (180° – A) + (180° – B) + (180° – C)

Simplifying the equation, we get:

Sum of Exterior Angles = 540° – (A + B + C)

Since the sum of interior angles of a triangle is always 180 degrees, we can substitute (A + B + C) with 180°:

Sum of Exterior Angles = 540° – 180°

Simplifying further, we find:

Sum of Exterior Angles = 360°

As we can see, the sum of exterior angles of a triangle is indeed 360 degrees, which aligns with our formula.

Example 2: Quadrilateral

Let’s move on to a quadrilateral, which has four sides. Extending each side will create four exterior angles. Let’s calculate their sum.

Consider a quadrilateral with angles A, B, C, and D. Extending side AB creates an exterior angle at D, extending side BC creates an exterior angle at A, extending side CD creates an exterior angle at B, and extending side DA creates an exterior angle at C.

Let’s measure these exterior angles:

  • Exterior Angle at A = 180° – A
  • Exterior Angle at B = 180° – B
  • Exterior Angle at C = 180° – C
  • Exterior Angle at D = 180° – D

Now, let’s calculate their sum:

Sum of Exterior Angles = (180° – A) + (180° – B) + (180° – C) + (180° – D)

Simplifying the equation, we get:

Sum of Exterior Angles = 720° – (A + B + C + D)

Since the sum of interior angles of a quadrilateral is always 360 degrees, we can substitute (A + B + C + D) with 360°:

Sum of Exterior Angles = 720° – 360°

Simplifying further, we find:

Sum of Exterior Angles = 360°

Once again, the sum of exterior angles of a quadrilateral is 360 degrees, confirming the validity of our formula.

Generalizing the Formula

From the examples above, we can observe that the sum of exterior angles of any polygon is always 360 degrees. This holds true for triangles, quadrilaterals, pentagons, hexagons, and any other polygon, regardless of the number of sides.

So, why does this formula work? To understand this, let’s consider the interior angles of a polygon. The sum of interior angles of an n-sided polygon can be calculated using the formula:

Sum of Interior Angles = (n – 2) * 180°

For example, a triangle has three sides, so its sum of interior angles is (3 – 2) * 180° = 180°. Similarly, a quadrilateral has four sides, so its sum of interior angles is (4 – 2) * 180° = 360°.

Now, if we subtract the sum of interior angles from 360 degrees, we get the sum of exterior angles:

Sum of Exterior Angles = 360° – Sum of Interior Angles

Substituting the formula for the sum of interior angles, we get:

Sum of Exterior Angles = 360° – (n – 2) * 180°

Simplifying further, we find:

Sum of Exterior Angles = 360° – 180°n + 360°

Combining like terms, we get:

Sum of Exterior Angles = 360° – 180°n + 360° = 720° – 180°n

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