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 the 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.

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.

The 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.

The 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.

The number of diagonals of rectangle = 2

The formula to find the length of the diagonal of a rectangle is:

Diagonal of a Rectangle = √[l² + b²]

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. 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 the 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, d₁ = p = √[2a²+2b² – q²]
• Diagonal, d₂ = q = √[2a²+2b² – p²]

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 the cube are the straight lines that pass through the centre of the cube and join the opposite vertices. The diagonals of the 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 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:

Length of Diagonals

The length of diagonals of different shapes depends on their dimensions.

Related Articles

• Square
• Rectangle
• Rhombus
• Parallelogram
• Area Of Square Using Diagonals
• Cube and Cuboid

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 50cm² and one of its diagonal measures is 7cm.

Solution:

Given:

Area of a rhombus = 50cm²

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.

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