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.
A pentagon has 5 straight sides and the shape must also be closed (all the lines should connect to each other):
Types of Pentagon
Regular Polygon
Irregular Polygon
Concave polygon
Convex Polygon
The General Rule
Each time we add a side (triangle to quadrilateral, quadrilateral to pentagon, etc), we add another 180° to the total sum.
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-
Properties of a Regular Pentagon
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
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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.
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)
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?
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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
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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?
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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.
The Pentagon and its Subtypes
A pentagon is a two-dimensional shape with 5 sides and 5 angles in geometry. An angle is produced in a pentagon when two of its sides share a common point. Because there are five vertices in a pentagon, there are five angles in a pentagon. In this article, we will go through the angles in a pentagon in-depth, including internal angles, exterior angles, the sum of angles in a pentagon, and so on, with numerous examples.
A pentagon is a two-dimensional closed polygon having five sides and five angles. A pentagon may be categorized into several varieties according to its qualities. They are as follows:
Regular Pentagon: A pentagon with equal sides and interior angles.
Irregular Pentagon: A pentagon's sides are not all equal, and the inner angles are not all the same size.
Convex Pentagon: All of the interior angles are less than 180 degrees, and all of the vertices point outwards. A convex pentagon is a regular pentagon.
Concave Pentagon: A concave polygon is formed when one of the inner angles of a pentagon is larger than 180° and one of the vertices points inward.
The diagrams below show the definitions of a regular pentagon, an irregular pentagon, and a concave pentagon.
The sum of a Pentagon's outside angles
We know that the formula for calculating the sum of a polygon's inner angles is (n – 2) 180°.
As a result, each interior angle = {(n – 2) 180°}/n.
Each exterior angle is known to be supplementary to the inner angle.
Thus, each outside angle = {180°n -180°n + 360°}/n = 360°/n may be calculated using the preceding method.
As a result, the sum of a polygon's exterior angles = n(360°/n).
Because a pentagon has five sides, n=5.
Interior View of a Standard Pentagon
The internal angles of a pentagon are the angles created by two consecutive pairs of sides.
The number of sides equals the number of vertices plus the number of internal angles, which equals five.
Neighboring angles or adjacent interior angles are two internal angles that share a common side.
All five sides of a regular pentagon are equal, as are all five angles. As a result, the formula below gives the measurement of each internal angle of a regular pentagon.
Each internal angle is measured as (n – 2) 180°/n = 540°/5 = 108°.
n = the number of sides in this case.
A Regular Pentagon's Exterior Angle
When the sides of a pentagon are stretched, the angles produced outside the pentagon with its sides are called exterior angles. A pentagon's outside angles are all equal to 72°.
Because the total of a regular pentagon's outside angles equals 360°, the formula for calculating each exterior angle of a regular pentagon is as follows:
Each exterior angle of a pentagon is measured as 360°/n = 360°/5 = 72°.
FAQs on Angles in a Pentagon
1. How Many Angles Does a Pentagon have and What is the Sum of the Internal Angles of a Pentagon?What is the General Rule?
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 a number of sides
Measure of each of the Angles in a Regular Polygon = 180 ×(n−2) degrees/n, where n is the number of sides.
2. What are the Properties of a Pentagon?
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.
3. What are the different types of Pentagons?
Pentagons are classified into several varieties based on their features. Here is a list of pentagon kinds classified by their sides, angles, and vertices:
When all of the sides and internal angles of a pentagon are equal, it becomes a regular pentagon. An irregular pentagon is one in which the sides are not equal and the angles are not of the same measure.
The vertices of a convex pentagon point outwards, and the inner angles are smaller than 180°. If any of the inner angles of a pentagon exceeds 180° and the vertices point inwards, the pentagon is concave.