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Understanding the Sum of Angles in a Polygon

CEO Khai Intela
Have you ever wondered how to calculate the sum of angles in a polygon? Well, it all depends on the number of edges and vertices. In this article, we will explore the different methods used...

Have you ever wondered how to calculate the sum of angles in a polygon? Well, it all depends on the number of edges and vertices. In this article, we will explore the different methods used to calculate the sum of interior angles and the sum of exterior angles of a polygon. So, let's dive in!

Types of Polygons

Before we get into the calculations, let's first understand the types of polygons. A polygon is considered regular when all its angles and sides are equal. There are two types of angles in a regular polygon:

  • Interior Angles: These angles lie inside the shape, typically a polygon.
  • Exterior Angles: An exterior angle of a polygon is the angle between a side and its adjacent extended side.

Sum of Interior Angles in a Polygon

The interior angles of a polygon are the angles that lie inside the shape. Take a look at the interior angles A, B, and C in the triangle below. In a regular polygon, all interior angles are equal to each other. To calculate the sum of interior angles of a polygon, we use the formula:

Sum of interior angles = (n − 2) × 180°

where 'n' represents the number of sides of the polygon.

Sum of Interior Angles of a polygon Image: Sum of Interior Angles of a polygon

Another way to calculate the sum of interior angles is by counting the number of triangles formed inside the polygon using the diagonals. Since the interior angles of a triangle sum up to 180°, we can multiply 180° with the number of triangles formed inside the polygon. For instance, a quadrilateral can be divided into two triangles using the diagonals. Therefore, the sum of the interior angles of a quadrilateral is 2 × 180° = 360°. Similarly, a pentagon can be divided into three triangles, so the pentagon's interior angles will sum up to 3 × 180° = 540°.

Let's take an example:

Example: What is the Sum of the Interior Angles in a Hexagon?

We know that a hexagon has 6 sides, so 'n' is equal to 6. Using the formula mentioned earlier, we can calculate the sum of the interior angles:

S = (n − 2) × 180
S = (6-2) × 180°
S = 4 × 180
S = 720°

Hence, the sum of the interior angles of a hexagon is 720°.

Sum of Exterior Angles in a Polygon

Now, let's focus on the sum of exterior angles in a polygon. An exterior angle of any shape or regular polygon is the angle formed by one side and the extension of the adjacent side of that polygon. Take a look at the exterior angles shown in the polygon below.

Sum of Exterior Angles of a polygon Image: Sum of Exterior Angles of a polygon

The sum of the exterior angles of a polygon is always equal to 360°. We can prove this by following these steps:

  1. We know that the sum of the interior angles of a regular polygon with 'n' sides is given by the formula 180°(n-2).
  2. The interior and exterior angles at each vertex form a linear pair. Therefore, there will be 'n' linear pairs in the polygon. Since each linear pair sums up to 180°, the sum of all linear pairs will be: 180n°.
  3. So, the sum of exterior angles = Sum of all linear pairs - Sum of interior angles.
  4. This can be simplified as: Sum of exterior angles = 180n - 180(n-2) = 180n - 180n + 360.
  5. Hence, the sum of exterior angles of a polygon equals 360°.

Related Articles

Now that you've gained a deeper understanding of the sum of angles in a polygon, you can apply these formulas to calculate and analyze various polygonal shapes. Remember, the sum of interior angles depends on the number of sides, while the sum of exterior angles is always 360°.

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