Sum of Angles in a Polygon

The sum of angles in a polygon depends on the number of vertices it has. As we know, polygons are closed figures, which are made up line-segments in a two-dimensional plane. There are different types of polygons based on the number of sides. They are:

  • Triangle (Three-sided polygon)
  • Square (Four-sided polygon)
  • Pentagon (Five-sided polygon)
  • Hexagon (Six-sided polygon)
  • Septagon (Seven-sided polygon)
  • Octagon (Eight-sided polygon)
  • Nonagon (Nine-sided polygon)
  • Decagon (Ten-sided polygon)

And so on.

Angle Sum of Polygons

As we know, by angle sum property of triangle, the sum of interior angles of a triangle is equal to 180 degrees. When we start with a polygon with four or more than four sides, we need to draw all the possible diagonals from one vertex. The polygon then is broken into several non-overlapping triangles.

Interior Angles Sum of Polygons

The angle sum of this polygon for interior angles can be determined on multiplying the number of triangles by 180°. After examining, we can see that the number of triangles is two less than the number of sides, always.

Hence, we can say now, if a convex polygon has n sides, then the sum of its interior angle is given by the following formula:

S = ( n − 2) × 180°

This is the angle sum of interior angles of a polygon.

Exterior Angles Sum of Polygons

An exterior angle of a polygon is made by extending only one of its sides, in the outward direction. The angle next to an interior angle, formed by extending the side of the polygon, is the exterior angle.

Hence, we can say, if a polygon is convex, then the sum of the degree measures of the exterior angles, one at each vertex, is 360°.

Therefore, the sum of exterior angles = 360°

Proof: For any closed structure, formed by sides and vertex, the sum of the exterior angles is always equal to the sum of linear pairs and sum of interior angles. Therefore,

S = 180n – 180(n-2)

S = 180n – 180n + 360

S = 360°

Also, the measure of each exterior angle of an equiangular polygon = 360°/n

Also, read:

How to find the sum of angles of a polygon?

Question 1: Find the sum of interior angles of a regular pentagon.

Solution: A pentagon has five sides.

Therefore, by the angle sum formula we know;

S = ( n − 2) × 180°

Here, n = 5

Hence,

Sum of angles of pentagon = ( 5 − 2) × 180°

S = 3 × 180°

S = 540°

Question 2: Find the measure of each interior angle of a regular decagon.

Solution: A decagon has ten sides.

Therefore, by the angle sum formula we know;

S = ( n − 2) × 180°

Here, n = 10

Hence,

Sum of angles of pentagon = ( 10 − 2) × 180°

S = 8 × 180°

S = 1440°

For a regular decagon, all the interior angles are equal.

Hence, the measure of each interior angle of regular decagon = sum of interior angles/number of sides

Interior angle = 1440/10 = 144°

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