Diagonals of a polygon are the lines that connect the alternate vertices of the polygon. A polygon of n sides has n(n-3)/2 diagonals. A polygon is a closed figure with n sides (where n is always greater than equal to 3). A polygon is a closed shape with three or more straight sides, and diagonals are the line segments that connect any two non-adjacent vertices of the polygon.
In this article, we’ll explore the concept of diagonals in polygons, examine their properties and patterns, and discuss their applications in various fields. So, let’s get started and dive into the exciting world of polygons and diagonals!
What are Polygons?
Polygon can be defined as a closed figure which is formed by joining the straight lines. So it is easy to see that to make a polygon at least three lines are needed. There are polygons known with different names depending on the number of lines by which it is formed. Some basic polygons are:
- Triangle: A polygon with three sides.
- Quadrilateral: A polygon with four sides.
- Pentagon: A polygon with five sides.
- Hexagon: A polygon with six sides.
- Heptagon: A polygon with seven sides.
- Octagon: A polygon with eight sides.
- Nonagon: A polygon with nine sides.
- Decagon: A polygon with 10 sides.
- Dodecagon: A polygon with 20 sides.
Definition of Diagonals
The diagonal of a polygon can be defined as a line that joins the end of two non-adjacent vertices of any polygon and it is generally inside a polygon for convex polygons and lies outside for concave polygons.
Number of Diagonals in a Polygon
As we know there are different polygons with each having a different number of sides, such as a triangle with three sides, quadrilateral with 4 sides, a pentagon with five sides, etc., and each polygon has a different number of diagonals.
- Triangle: There are no diagonals for a triangle.
- Quadrilateral: There are two diagonals for a quadrilateral.
- Pentagon: There are five diagonals for a pentagon.
- Hexagon: There are nine diagonals for a hexagon.
Formula for Diagonal of Polygon
The formula for the number of diagonals of a polygon is given as follows:
Diagonals = (n × (n – 3))/2
Where n is the number of sides of a polygon
Proof:
For making a diagonal in a polygon we need two vertices. Let’s consider an N-sided polygon, now each vertex can be connected to the other in nC2 different ways but in this, the number of sides which is n is taken twice so subtract n from the total number of ways.
Hence number of diagonals = nC2 – n
= (n!)/(n – 2)! × (2!) – n
= n(n – 1)(n – 2)!/(n – 2)! × (2!) – n
= n(n – 1)/2 – n
= (n(n – 1) – 2n)/2
= n(n – 3)/2
Examples of Calculating the Number of Diagonals in a Polygon
Example 1: How many diagonals does a Triangle have?
Solution:
As triangle has 3 sides.
So, for triangle n = 3
Using the formula, diagonals = (n × (n – 3))/2
Diagonals = (3 × (3 – 3))/2
⇒ Diagonals = 0
Hence, a triangle has zero diagonals.
Example 2: Find the number of diagonals of a Square or any other quadrilateral.
Solution:
As Square or any other quadrilateral has 4 sides.
So, for square n = 4
Using formula, diagonals = (n×(n-3))/2
Put n = 4
Diagonals = (4 × (4 – 3))/2
⇒ Diagonals = 2
Hence, a square or any other quadrilateral has two diagonals.
Example 3: How many diagonals does a Pentagon have?
Solution:
As a pentagon has 5 sides.
So, for pentagon n = 5
Using the formula, diagonals = (n × (n – 3))/2
Put n = 5
Diagonals = (5 × (5 – 3))/2
⇒ Diagonals = 5
Hence, a pentagon has five diagonals.
Properties for Diagonals of a Polygon
There are various properties diagonals are associated with for various different polygons such as different types of quadrilaterals and regular polygons. Some of the Properties of diagonals with various different polygons are as follows:
For Square
- The diagonal of a square bisects the square into two congruent right triangles.
- The length of the diagonal of a square can be found using the Pythagorean theorem: d = √2×s, where d is the length of the diagonal and s is the length of one of the sides of the square.
- The diagonal of a square is also a line of symmetry for the square.
For Parallelogram
- The diagonal of a parallelogram divides the parallelogram into two congruent triangles.
- The diagonal of a parallelogram bisect each other.
- If diagonals are equal to a parallelogram then it is a rectangle.
For Rhombus
- Diagonals of a rhombus are perpendicular bisectors of each other and also divide the rhombus into two triangles with equal area.
For Regular Polygon
- For regular polygons with an even number of sides, diagonals joining the opposite vertices intersect at a point which is called the center of the polygon.
- For regular polygons with an odd number of sides, diagonals joining the opposite vertices don’t intersect at the center of the polygon.
Diagonals in Convex and Concave Polygons
Convex and concave polygons are defined based on the position of diagonals. If all the diagonals of a polygon lie inside of the area bounded by its side, then it is called a convex polygon whereas if any one of the diagonals of a polygon lies outside of the area bounded by its side, then it is called a concave polygon.
One another definition of Convex and Concave Polygons includes interior angles. If all internal angles of a polygon are strictly less than 180° then it is called a convex polygon while if any of the interior angles is strictly greater than 180° then it is called a concave polygon.
Lengths of Diagonals in Regular Polygons
As regular polygons equal sides and interior angles, we can find the formula for the length of regular polygons. The formula for the length of the diagonal of a regular polygon is given as:
Where,
- d is the length of diagonal,
- s is the length of the side, and
- n is the number of sides of the polygon.
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Sample Problems on Diagonals of a Polygon
Problem 1: How many diagonals does a hexagon have, find using the diagonal of a polygon formula.
Solution:
Hexagon is a polygon that is formed by six straight lines
So a hexagon has 6 sides so n = 6
Using formula, diagonals = (n × (n – 3))/2
Put n = 6
Diagonals = (6 × (6 – 3))/2
⇒ Diagonals = 9
Hence a hexagon has nine diagonals.
Problem 2: There are 20 diagonals in a polygon, find the number of sides it has.
Solution:
Using diagonals formula = (n × (n – 3))/2
So 20 = (n × (n – 3))/2
⇒ 20 × 2 = (n × (n – 3))
⇒ 40 = n2 – 3 × n
⇒ n2 – 3 × n – 40 = 0
⇒ n2 – 8n + 5n – 40 =0
⇒ n(n – 8) + 5(n- 8) = 0
⇒ (n – 8)(n + 5) = 0
So, n = 8
Hence the polygon is the octagon (polygon with 8 sides).
Problem 3: How many diagonals do a decagon has, find using the diagonal of a polygon formula.
Solution:
A decagon has 10 sides so n = 10
Using formula, diagonals = (n × (n – 3))/2
Put n = 10
Diagonals = (10 × (10 – 3))/2
= 35
Hence a decagon has 35 diagonals.
Problem 4: There are 27 diagonals in a polygon, find the number of sides it has.
Solution:
Using diagonals formula = (n × (n – 3))/2
So 27 = (n × (n – 3))/2
⇒ 27 × 2 = (n × (n – 3))
⇒ 54 = n2 – 3 × n
⇒ n2 – 3 × n – 54 = 0
⇒ n2 – 9n + 6n – 54 =0
⇒ n(n – 9) + 6(n- 9) = 0
⇒ (n – 9)(n + 6) = 0
So, n = 9
Hence the polygon is Nonagon (polygon with 9 sides).
Problem 5: How many diagonals does a polygon have if the sides are 20?
Solution:
Put n = 20 in diagonals formula
Diagonals = (20 × (20 – 3))/2
⇒ Diagonals = 170
Hence there will be 170 diagonals in a 20 sided polygon.
Problem 6: There are 405 diagonals in a polygon, find the number of sides it has.
Solution:
Using diagonals formula = (n × (n – 3))/2
So 405 = (n × (n – 3))/2
⇒ 405 × 2 = (n × (n – 3))
⇒ 810 = n2 – 3 × n
⇒ n2 – 3 × n – 810 = 0
⇒ n2 – 30n + 27n – 810 =0
⇒ n(n – 30) + 27(n – 30) = 0
⇒ (n – 30)(n + 27) = 0
So, n = 30
Hence the polygon has 30 sides.
Problem 7: How many diagonals does a polygon have if the sides are 40?
Solution:
Put n = 40 in diagonals formula
Diagonals = (40 × (40 – 3))/2
⇒ Diagonals = 740
Hence, there will be 740 diagonals in a 40 sided polygon.
FAQs on Diagonals of a Polygon
Q1: What is a Diagonal in a Polygon?
Answer:
Diagonal is the straight line joining the two non-adjacent vertices of any polygon.
Q2: How many Diagonals does a Polygon have?
Answer:
A polygon can have a maximum of n(n-2) diagonals, where n is the number of sides in the polygon.
Q3: How do you Calculate the Number of Diagonals in a Polygon?
Answer:
We can calculate the number of diagonals any polygon can have using the following formula:
Number of Diagonals of Polygon = n(n-2)
Where n is the number of sides of the polygon.
Q4: What are the Properties of Diagonals in Polygons?
Answer:
Diagonal for different geometric structures hold different properties, some of these properties are:
- For Square
- The diagonal of a square bisects the square into two congruent right triangles.
- The diagonal of a square is also a line of symmetry for the square.
- For Parallelogram
- The diagonal of a parallelogram divides the parallelogram into two congruent triangles.
- If diagonals are equal for a parallelogram then it is a rectangle.
Q5: How do you Construct Diagonals in Polygons?
Answer:
To construct a diagonal of any polygon we choose any two non-adjacent vertices of that polygon and draw a line segment joining these two vertices. The drawn line is the diagonal of the given polygon.