 # Angles in a Pentagon

What is a Pentagon?

• In simple mathematics, a polygon can be defined as any 2-dimensional shape that is formed with straight lines.

• In the case of quadrilaterals or triangles and pentagons, they are all perfect examples of polygons.

• The interesting aspect is that the name of any kind of polygon highlights the number of sides the polygon possesses.

Is it a Pentagon?

• A pentagon has 5 straight sides and the shape must also be closed (all the lines should connect to each other):

Types of Pentagon-

1. Regular Polygon

2. Irregular Polygon

3. Concave polygon

4. Convex Polygon

The General Rule

What is a Regular Polygon?

If all the sides of a polygon are equal and all the angles of a polygon are equal then the polygon is known as a regular polygon.

Different Shapes, Number of Sides, Sum of Interior Angles and the Measure of each Angle-

 Shape Sides Sum of Interior Angles Each Angle Triangle 3 sides 180 degrees 60 degrees Quadrilateral 4 sides 360 degrees 90 degrees Pentagon 5 sides 540 degrees 108 degrees Hexagon 6 sides 720 degrees 120 degrees Heptagon ( or Septagon) 7 sides 900 degrees 128.57 degrees Octagon 8 sides 1080 degrees 135 degrees Nonagon 9 sides 1260 degrees 140 degrees

 General Rule Sum of Interior Angles of a polygon = 180 ×(n−2)  degrees, where n is number of sides Measure of each of the Angle (in a Regular Polygon) = 180 degrees ×(n−2)  / n, where n is the number of sides/.

Properties

A regular pentagon has the following properties:

• Interior angles that measure 108°

• Exterior angles that measure 72°

• A regular pentagon has an area of approximately 1.7204774 × s2 (where s is equal to the side length)

Any pentagon has the following properties:

• Sum of Interior Angles of measure 540°

• Number of diagonals is five.

Sum of Angles in a Pentagon:

To find the sum of the angles in a pentagon, divide the pentagon into different triangles. There are three angles in a triangle. Since the sum of the angles of the triangles is equal to 180 degrees.

3 x 180 = 540 degrees

Therefore, the sum of angles in a pentagon is 540 degrees.

Regular Pentagons:

The properties of Regular Pentagons-

1. All the sides are of the same length (congruent, that is they are equal) and all the interior angles are of the same size (congruent).

2. Now to find the measure of the interior angles of the pentagon, we know that the sum of all the angles in a pentagon is equal to 540 degrees (from the above figure)and there are five angles. (540/5 = 108 degrees)

3. So, the measure of the interior angle of a regular pentagon is equal to 108 degrees.

How to Measure the Central Angles of a Regular Pentagon?

To find the measure of the central angle of a regular pentagon,we need to make a circle in the middle of the pentagon.We know that a circle is 360 degrees around. Now ,divide that by five angles. Now the measure of each central angle is equal to 360/5 = 72 degrees.

So, the measure of the central angle of a regular pentagon is equal to 72 degrees.

Pentagon with Right Angle

• A pentagon has five sides.

• A regular pentagon has no right angles ( It has interior angles each equal to 108 degrees).

• An irregular pentagon has at most three right angles, because a fourth would leave 180 degrees to be used for the final angle that is (540 degrees - 360 degrees), which is a straight line.

• For a pentagon to have any right angles however, the pentagon has to be uneven,since the sum of all interior angles of a pentagon always sums up to 540 degrees.

Convex  Pentagon and Concave Pentagon

If all the vertices of a pentagon are pointing outwards, then the pentagon can be known as a convex pentagon. If a pentagon has at least one vertex pointing inside, then the pentagon can be known as a concave pentagon.

Questions to be Solved-

Question 1) Is the Diagram Given Below a Pentagon?

Answer) The figure given below cannot be known as a pentagon because from the properties of a pentagon we know that it should be a closed figure. But the figure given above is open therefore, it is not a pentagon.

Q1. How Many Angles Does a Pentagon have and What is the Sum of the Internal Angles of a Pentagon?What is the General Rule?

Ans. As the name suggests, a pentagon has 5 straight sides . In geometry, a pentagon (from the Greek  pente and  gonia, which means five and angle) is any five-sided polygon or also known as 5-gon. The sum of the internal angles in a simple pentagon is always equal to 540°. Here’s the general rule-

Sum of the Interior Angles of a polygon = 180 × (n−2)  degrees, where n is number of sides

Measure of each of the Angle in a Regular Polygon = 180  ×(n−2)  degrees/n, where n is the number of sides.

Q2. What are the Properties of a Pentagon?

Ans. Here are some of the properties of the pentagon -

• In the pentagon, the sum of the interior angles is always equal to 540 degrees.

• If all the sides of the pentagon are equal and all the angles of the pentagon are of equal measure, then it is known as a regular polygon. Otherwise, it is known as an irregular polygon.

• In the case of a regular pentagon, the interior angle is equal to 108°, and the exterior angle is  equal to 72°.

• An equilateral pentagon has five sides that are equal to each other.

• The sum of the interior angles of a rectangular pentagon is equal to 540 degrees.