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A diagonal is a line segment, it is a straight line connecting two opposites of a polygon of any shape. A polygon is a geometrical closed figure that has a finite number of sides and angles. It is made up of straight lines connected to each end. Triangle, Rectangle, Square, Pentagon, Hexagon, are some examples of polygons. A Diagonal in any polygon is that line which is connected from the vertex to a non-adjacent vertex. Hence, the triangle won’t have a diagonal, we cannot draw a line from one interior angle to any other interior angle that is not also a side of the triangle.
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Key Takeaways: Diagonal, Triangle, Rectangle, Square, Pentagon, Hexagon
What is a Diagonal?
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A diagonal is a line segment that joins one corner of a polygon to another i.e., it joins two non-adjacent vertices of a polygon opposite to each other. A diagonal is a straight line that connects the opposite corners of a polygon through its vertices i.e., the diagonal of a polygon is a line segment that joins any two non-adjacent corners. Different polygons may have a different number of diagonals depending on the number of sides of the polygon. Since a diagonal is a line segment joining non-adjacent vertices, the shape of a diagonal is a straight line.
Diagonals of polygons - Formula
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To find the number of diagonals in a polygon, we can use the formula given below. We can easily find the number of diagonals in a polygon if we know the number of sides of the polygon.
| Number of Diagonals of Polygon = \({n(n-3) \over 2}\) |
Where n refers to the number of sides of a polygon. With this formula, we can easily calculate the number of diagonals in a given polygon.
Solved Examples
We can find Diagonals of any shape as we have done in following examples:
- Number of Diagonals in Triangle
A Triangle has 3 sides,
hence the number of diagonals = 3(3-3)/2 =0.
Therefore, the triangle has 0 diagonals.
- Number of Diagonals in Rectangle
A Rectangle has 4 sides,
hence the number of diagonals = 4(4-3)/2 =2.
Therefore, the Rectangle has 2 diagonals.
- Number of Diagonals in Pentagon
A Pentagon has 5 sides,
hence the number of diagonals = 5(5-3)/2 =5.
Therefore, the Pentagon has 5 diagonals.
The table given below shows the number of diagonals for some polygons:
| Shape of Polygon | No. of sides (n) | No. of Diagonals |
|---|---|---|
| Triangle | 3 | 3(3−3)/2 = 0 |
| Quadrilateral | 4 | 4(4−3)/2 = 2 |
| Pentagon | 5 | 5(5−3)/2 = 5 |
| Hexagon | 6 | 6(6−3)/2 = 9 |
| Heptagon | 7 | 7(7−3)/2 = 14 |
| Octagon | 8 | 8(8−3)/2 = 20 |
| Nonagon | 9 | 9(9−3)/2 = 27 |
| Decagon | 10 | 10(10−3)/2 = 35 |
| Hendecagon | 11 | 11(11−3)/2 = 44 |
| Dodecagon | 12 | 12(12−3)/2 = 54 |
Diagonals of polygons - Formula
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The length of a diagonal for any polygon depends upon the type of the polygon. There is no general formula to calculate the length of the diagonal of polygons. The formula to calculate the length of the diagonal can be derived, based on the dimensions of the particular polygon.
- Formula of Length of Diagonal of Square = s√2
Where s refers to the length of the side of the square.
- Formula of Length of Diagonal of Rectangle = √ (a2+b2)
Where a refers to the length of the rectangle.
b refers to the breadth of the rectangle.
p and q are the diagonals.
- Formula of Length of Diagonal of Parallelogram
The formula of the diagonal of parallelogram in terms of sides and cosine β (cosine theorem) if x = d1 and y = d2 are the diagonals of a parallelogram and a and b are the two sides of a parallelogram.
x = d1 = √ (a2+b2−2abcosβ)
y = d2 = √ (a2+b2+2abcosβ)
The formula of parallelogram diagonal length in terms of cosine α
x = d1 = √ (a2+b2+2abcosα)
y = d2 = √ (a2+b2-2abcosα)
The formula of parallelogram diagonal length in terms of two sides and other diagonal
x = d1 = √(2a2+2b2−d22)
y = d2 = √(2a2+2b2−d12)
- Formula of Length of Diagonal of a Cube = s √3
Where s refers to the side of a cube. We use the Pythagoras theorem to find the diagonal of the cube.
- Formula of Length of Diagonal of Cuboid = √ (l2+b2+h2)
Where l, b, h refers to the length, breadth, and height of a cuboid.
Why Do We Use Diagonals?
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Diagonals always need a base for any structure. We have to always put rods diagonally in a cleared frame so that it cannot break. There are various big rods which are placed in a diagonal position in very high buildings so that the buildings cannot fall or crash. We can use diagonals in everyday life.
Points to Remember
Following are some important points:
- A diagonal of a polygon is a line from the vertex to a non-adjacent vertex.
- The number of diagonals in a polygon depends upon the number of sides of the polygons.
- Formula to find the Number of Diagonals of a Polygon = n(n−3)/2.
- The triangle has 0 diagonals.
- The length of the diagonal for any polygon depends upon the type of the polygon.
Sample Questions
Ques: What is the formula to find the number of diagonals in a Polygon? (2 Marks)
Ans: The number of diagonals in a polygon depends upon the number of sides of the polygons.
Number of Diagonals of a Polygon=n(n−3)/2,
where n refers to the number of sides of a polygon.
If n = 4,
Diagonals are n (n-3)/2
=> 4 (4-3)/2
=> 4*1/2
=> 4/2
=> 2
Ques: Can a triangle have a diagonal? Explain with a numerical example? (2 Marks)
Ans: A Triangle has 3 sides,
The formula to find the number of diagonals is n(n-3)/2.
Therefore, the number of diagonals in a triangle = 3(3-3)/2 =0.
Ques: What is the formula to find the length of a diagonal in a cuboid? Mention an example? (2 Marks)
Ans: Length of Diagonal of Cuboid Formula = √ (l2+b2+h2).
Where ‘l’ is the length, ‘b’ is the breadth, and ‘h’ is the height of a cuboid.
Example: Let there be a cuboid of 3x4x5cm as its dimension, then the length of the diagonal of the cuboid = √ (32+42+52) = 7.07cm.
Ques: Find the number of diagonals in a polygon with 7 sides? (2 Marks)
Ans: Given, the polygon has 7 sides i.e., n=7
The formula to find the number of diagonals is n(n-3)/2.
Therefore, the number of diagonals in the polygon with 7 sides is = 7(7-3)/2=14.
Ques: Find the length of a diagonal of a cube with a side of 5 cm? (2 Marks)
Ans: The formula to find the length of the diagonal in cube = s√3.
Given a = 5cm,
Therefore, the length of the diagonal of the cube = 5√3 = 8.66 cm.
Ques: Find the number of diagonals in a polygon with 8 sides? (2 Marks)
Ans: The formula to Formula to find the number of diagonals is n(n-3)/2
Given number of sides i.e., n = 8cm,
Therefore, the number of diagonals in a polygon with 8 sides = 8(8-3)/2=10.
Ques: Find the length of a diagonal of a Rectangle with 7 x 8 cm as its sides?(2 Marks)
Ans: The formula to find the length of a diagonal of Rectangle = √ (a2+b2)
Given a=7 cm and b=8cm,
Therefore, the length of a diagonal of Rectangle = √ (72+82)
= 10.6 cm.
Ques: Find the length of a diagonal of the Rectangle with 3 x 8 cm as its sides?(2 Marks)
Ans: The formula to find the length of a diagonal of Rectangle = √ (a2+b2),
Given a = 3cm and b= 8cm,
Therefore, the length of a diagonal of Rectangle = √ (32+82) = 8.54 cm.
Ques: What is the formula to find the length of the diagonal in the polygon? (1 Mark)
Ans: There is no particular formula to find the length of the diagonal in a polygon. The length of a polygon depends upon the type of polygon.
However, we can find the number of diagonals of a quadrilateral such as a square.
As square has 4 sides, then n = 4,
Diagonals are n (n-3)/2
=> 4 (4-3)/2
=> 4*1/2
=> 4/2
=> 2
Ques: The area of a rhombus is 50 cm2, one of its diagonal measures 7cm then determines the other diagonal of the rhombus?(2 Marks)
Ans: Given that Area of the rhombus is A= 50cm2,
Let's say one of the given diagonals is x = 7cm,
Now, we need to find another diagonal that says ‘y’.
The formula to find the diagonal, y is given as = 2(A)/x
= 2(50)/7
= 100/7
= 14.28.
Hence, the other diagonal of the rhombus is approximately 14.3 cm.
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