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How Can You Calculate the Angles Of a Pentagon?

A pentagon is one of the types of polygons. It is a geometrical shape with five sides and five angles. The word ‘pentagon’ is derived from the Greek word ‘pente’ meaning five and ‘gonia’ meaning angle. All the five sides of a pentagon meet each other to form a two-dimensional shape. 

This article will learn about the pentagon, its properties, and calculating different angles in a pentagon formula. 

Polygons and types of polygons

Our geometry coursebooks contain studies of various two-dimensional shapes, and polygon is one of them. A polygon is any figure made up of more than two straight lines which meet each other to form a shape, and the angles between the sides are called interior angles. We can define a polygon by the number of sides it has, for example:

  • A triangle is a three-sided polygon
  • The quadrilateral is a four-sided polygon
  • Pentagon is a five-sided polygon
  • Hexagon is a six-sided polygon
  • Heptagon is a seven-sided polygon
  • Octagon is an eight-sided polygon

We will learn more about the pentagon and its properties in this article.

Shapes of a Pentagon

A pentagon has five sides; however, not all sides need to be of equal length. Hence, a pentagon can be a regular pentagon or an irregular pentagon, depending on the measurements of the sides. 

The two main types of the pentagon are:

  • Regular and Irregular

The regular pentagon has equal measurements of sides, and the angles between the sides are also equal. If the measures of the sides are not equal, then the angles formed between the sides are also unequal; this kind of pentagon is an irregular pentagon.

  • Convex and Concave

Identifying concave and convex pentagons is easy. If any one of the vertices is pointing inward, then it is a concave pentagon. A convex pentagon has all the vertices pointing outwards. 

Important Notes on Pentagon Angles

Before one begins learning about how to calculate the angles of a pentagon, one must be acquainted with the basic properties of a pentagon, which establishes the foundation of further calculations:

  1. A pentagon has five sides for calculating five interior angles.
  1. The five straight lines create vertices but do not intersect each other.
  2. The total sum of the interior angles of a pentagon equates to 540°.

Angles of a Pentagon 

  • Interior Angles

In a pentagon, the angle between the two joined sides is called an interior angle. The number of sides equals the number of angles in any polygon so that a pentagon would have five interior angles. 

If the pentagon is a regular pentagon, meaning all the sides are of equal measure, then all interior angles will have equal angles. The basic formula to calculate each interior angle is [(n – 2) × 180°]/n; here, ‘n’ is the number of sides, which is ‘n’ = 5. Putting the formula’s ‘n’ value, we get [(5 – 2) × 180°]/5 = 108°. This means, for a regular pentagon, the value of each interior angle equates to 108°. 

In an irregular pentagon where the sides are not equal to each other, there is one point to keep in mind: the sum of all the interior angles of any given pentagon always adds up to 540°. 

Usually, the question is to find the missing angle when the other four angles are given/known. Suppose the missing angle is ‘e’, the formula is quite simple: e° = [540°- (a°+  b° + c° + d°)]. The first step is to add all the four known angles = a°  +b° + c°  +d°, then subtract the total sum from 540°. 

  • Exterior Angles

An exterior angle of a pentagon is formed when each side of the pentagon is extended outward. The angle formed corresponding to the side is known as an exterior angle. 

All the exterior angles add up to 360° of a regular pentagon. Since the sides of a regular pentagon are equal, each exterior angle equates to 72°. The basic formula to calculate the exterior angle of a regular pentagon is [360°/n]; ‘n’ being the number of sides which is ‘5’ in this case : [360°/5] =  72°.

Another way to determine the exterior angles is to shrink the extended sides of a polygon to a point. The angle around any given point is always 360°, which is the sum of the exterior angles of a pentagon.

Total Sum of Interior Angles in a Pentagon

The sum of interior angles of a pentagon is always 540°. To know why mathematically it adds up to 540°, we divide a pentagon into three different triangles by drawing lines from one corner of the figure to another. 

The sum of angles in any given triangle is 180°, so for three triangles, we multiply the sum with the total number of triangles which is three: 180° × 3 = 540°. 

Hence, the sum of all interior angles of a pentagon equals  540°.

Properties and Examples of a Pentagon 

  1. A  regular pentagon contains ten isosceles right triangles.
  1. The diagonals of a regular pentagon can be calculated using n × (n ? 3) ÷ 2 = 5 × (5 ? 3) ÷ 2 = 5 (n = number of sides in a pentagon)
  1. The sum of all interior angles of a regular pentagon is 540°, and all exterior angles are 360°.
  1. Each interior angle equals 108°, and each exterior angle equals 72° of a regular pentagon. In the case of an irregular pentagon, the missing angle can be calculated based on known angels.
  1. If a pentagon is convex, the interior angles are not greater than 180°. All regular pentagons are convex pentagons.
  1. One of the vertices points inward for a concave pentagon, making one of the interior angles of a concave pentagon greater than 180°.

Conclusion

The geometric shapes we study always have real-life examples; for instance, the pentagon shape can be identified in the black sections in a soccer ball, the Pentagon building in the US, traffic crossing signs, home plate in baseball, etc. This shape can also be spotted in some flowers. Observation becomes easy once the properties are clarified and the uses are understood.

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