Diagonals
Diagonals are the parts of a shape, in geometry. In Mathematics, a diagonal is a line that connects two vertices of a polygon or a solid, whose vertices are not on the same edge. In general, a diagonal is defined as a sloping line or the slant line, that connects to vertices of a shape. Diagonals are defined as lateral shapes that have sides/edges and corners. We can find the diagonals for curved shapes, such as circles, spheres, cones, etc.
The word diagonal is derived from the Greek word “diagonios” which means “from angle to angle”. Also, in matrix algebra, the diagonal of the square matrix defines the set of entities from one corner to the farthest corner. In this article, let us discuss the meaning of the diagonal line, diagonals for different polygons such as square, rectangle, rhombus, parallelogram, etc. with its formulas.
Table of Contents: |
What are Diagonals?
A diagonal line is a line segment that connects the two vertices of a shape, which are not already joined by an edge. It does not go straight up, down or across. The shape of the diagonals is always a straight line.
In other words, a diagonal is a straight line that connects the opposite corners of a polygon or a polyhedron, through its vertex.
An example of diagonals is shown below.
In the above figure, AC and AC’ are the diagonals of the shape.
Diagonals Formula
If “n” is the number of vertices of a polygon, then the number of diagonals of a polygon can be found using the formula:
Number of diagonals of a polygon with “n” vertices = [n(n-3)]/2
Let us take an example of a square. A square has 4 vertices. Now, use the above formula to find the number of diagonals of a square.
Number of the diagonals of square = 4(4-3)/2
= 4(1)/2 = 2
Thus, the number of diagonals of the square is 2.
Diagonals of Shapes
Different shapes have different numbers of diagonals of different lengths. Let us discuss here the diagonals of various shapes in geometry.
- Diagonals of Triangle
- Diagonals of Square
- Diagonals of Rectangle
- Diagonals of Rhombus
- Diagonals of Parallelogram
- Diagonals of Pentagon
- Diagonals of Hexagon
- Diagonals of Cube
- Diagonals of Cuboid
Diagonals of Triangle
A triangle is a three-sided enclosed polygon, that has three vertices. No two vertices of the triangle are non-adjacents. Hence, a triangle does not have any diagonal.
Number of diagonals for a triangle = 0
Diagonals of Square
The diagonals of a square are the line segments that link opposite vertices of the square. A square has two diagonals. The two diagonals of the square are congruent to each other. The diagonals of a square bisect each other. Each diagonal cuts the square into two congruent isosceles right triangles.
Number of diagonals of square = 2
The formula to find the length of the diagonal of a square is:
Diagonal of a Square = a√2
Where “a” is the length of any side of a square.
Diagonals of Rectangle
A rectangle has two diagonals as it has four sides. Like a square, the diagonals of a rectangle are congruent to each other and bisect each other. If a diagonal bisects a rectangle, two congruent right triangles are obtained.
Number of diagonals of rectangle = 2
The formula to find the length of the diagonal of a rectangle is:
Diagonal of a Rectangle = √[l^{2} + b^{2}]
Where “l” and “b” are the length and breadth of the rectangle, respectively.
Diagonals of Rhombus
A rhombus has four sides and its two diagonals bisect each other at right angles. If all the angles of a rhombus are 90 degrees, a rhombus is a square or a rectangle. Since all the sides of the rhombus are congruent, and the opposite angles are parallel to each other, the area of the rhombus is given as:
Area of a rhombus, A = (½) pq square units
Where “p” and “q” are the two diagonals of the rhombus.
From the formula of area of a rhombus, we can easily find the diagonal of the rhombus.
Thus, the formula to find the length of the diagonal of the rhombus is:
Diagonal of a Rhombus, p = 2(A)/q and q = 2(A)/p
Diagonals of Parallelogram
A parallelogram is a quadrilateral. The opposite sides and angles of a parallelogram are congruent, and the diagonals bisect each other. The length of the diagonals of the parallelogram is determined using the formula:
Diagonal of a parallelogram:
- Diagonal, d_{1} = p = √[2a^{2}+2b^{2} – q^{2}]
- Diagonal, d_{2} = q = √[2a^{2}+2b^{2} – p^{2}]
Diagonals of Pentagon
A pentagon is a five-sided closed shape or a polygon, that has five vertices. A regular polygon has all its sides equal in length. A pentagon has a total of five diagonals that are joined through opposite and non-adjacent vertices.
Diagonals of pentagon = 5
Diagonals of Hexagon
A hexagon is a six-sided closed shape, that has five vertices. It is a polygon, that has a total of nine diagonals when the non-adjacent corners are joined together.
Diagonals of Hexagon = 9
Diagonals of Cube
A cube is a three-dimensional shape, that has six square faces of equal dimensions. It has 12 edges and 8 vertices. The primary diagonals of cube are the straight lines that pass through the center of cube and join the opposite vertices. The diagonals of faces of the cube, are the straight lines that join the opposite vertices on each face.
Hence,
- Number of primary diagonals of cube = 4
- Number of diagonals on the faces of cube = 12
- Total diagonals of the cube = 12 + 4 = 16
Diagonals of Cuboid
A cuboid is also a three-dimensional shape, that has six rectangular faces. Similar to the cube, it has 12 edges and 8 vertices. It is also called a rectangular prism. Since the structure of cube and cuboid are similar, therefore, the number of diagonals of both the shape will also be equal.
Hence,
- Number of primary diagonals of cuboid = 4
- Number of diagonals on the faces of cuboid = 12
- Total diagonals of the cuboid = 12 + 4 = 16
Number of Diagonals in Polygons
The following table gives the summary of the number of diagonals of different polygons:
Polygons | Number of vertices | Calculation | Number 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 |
Length of Diagonals
The length of diagonals of different shapes depends on their dimensions.
Length of Diagonal | Formulas |
Length of diagonal of Square | a√2
where a is the length of side of square |
Length of diagonal of Rectangle | √(l^{2 }+ b^{2})
Where, l and b are the length and breadth of the rectangle, respectively |
Length of diagonal of Cube | a√3
Where a is the length of edges of the cube |
Length of diagonal of Cuboid | √(l^{2 }+ b^{2 }+ h^{2})
Where l, b and h are length, breadth and height of cuboid |
Related Articles
Diagonals Solved Examples
Go through the below problems to find the diagonal of a polygon.
Example 1:
Find the diagonal of a square whose side measure is 4cm
Solution:
Given:
Side, a = 4 cm
We know that the formula to find the diagonal of a square is:
Diagonal of a Square = a√2
Now, substitute the side value, we get:
Diagonal, d = 4√2
Now, put the value of √2, which is equal to 1.414
d= 4 × 1.414
d = 5.656.
Thus, the diagonal of a square is 5.656 cm.
Example 2:
Determine the diagonal of a rhombus whose area is 50cm^{2} and one of its diagonal measures is 7cm.
Solution:
Given:
Area of a rhombus = 50cm^{2}
One of the diagonal of a rhombus, say q = 7cm
Thus, the formula to find the diagonal, p is given as:
Diagonal, p = 2(A)/q
Now, substitute the given values in the formula, we get:
p = 2(50)/7
p = 100/7
p= 14.28, which is approximately equal to 14.3
Hence, the other diagonal of a rhombus is 14.3 cm.
Register with BYJU’S -The Learning App and download the app to learn all the important Maths-related articles and formulas with ease.
Frequently Asked Questions on Diagonals
What is diagonal?
A diagonal is a straight line that connects the opposite corners of the polygon through its vertex.
What is the formula to calculate the diagonal of a polygon?
If “n” is the number of vertices of a polygon, the formula to calculate the number of diagonals of a polygon is [n(n-3)]/2
What is the length of the diagonal of a square?
The length of the diagonal of the square is a√2, where “a” is the length of any side of a square.
How many diagonals does a heptagon have?
A heptagon has 14 diagonals, as it has 7 vertices.
Does a circle have a diagonal?
No, a circle does not have a diagonal, as it has no vertices and sides.