What is Diagonal?
We have studied what a polygon is. To recall, a two-dimensional closed figure bounded with three or more than three straight lines is called a polygon. Triangles, square, rectangle, pentagon, hexagon, are some examples of polygons.
The segments are referred to as the sides of the polygon. The points at which the segments meet are called vertices. Segments that share a vertex are called adjacent sides. A segment whose endpoints are nonadjacent vertices is called a diagonal.
What is Diagonal?
A diagonal of a polygon is a line from a vertex to a non-adjacent vertex. So a triangle has no diagonals. You cannot draw a line from one interior angle to any other interior angle that is not also a side of the triangle. A quadrilateral, the next-simplest, has two diagonals. A pentagon has five diagonals. The shapes with a diagonal drawn on them are given below:
(Image will be uploaded soon)
We have a number of diagonals formula of a polygon which help us to know the number of diagonals in a polygon that is
Where the number of sides of a polygon is n
From this formula, we can easily calculate the number of diagonals in a polygon.
The Given Table Shows the Number of Diagonals in Different Polygons
Let us study diagonal formula for different shapes
Diagonal of Square Formula
\[\text{Diagonal of Square Formula} = s \sqrt{2}\]
Where, s represents the length of the side of the square
Diagonal of Rectangle Formula
\[\text{Diagonal of Rectangle Formula} = \sqrt{(a^{2} + b^{2})}\]
Where,
a is the length of the rectangle.
b is the breadth of the rectangle.
p and q are the diagonals.
Diagonal of Parallelogram Formula
The formula of parallelogram diagonal 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.
\[x = d_{1} = \sqrt{(a^{2} + b^{2} - 2 ab cos \beta})\]
\[y = d_{2} = \sqrt{(a^{2} + b^{2} + 2 ab cos \beta})\]
The formula of parallelogram diagonal in terms of sides and cosine α (cosine theorem)
\[x = d_{1} = \sqrt{(a^{2} + b^{2} + 2 ab cos \alpha})\]
\[y = d_{2} = \sqrt{(a^{2} + b^{2} - 2 ab cos \alpha})\]
Formula of diagonal of parallelogram in terms of two sides and other diagonal
\[x = d_{1} = (\sqrt{2a^{2} + 2b^{2} - d_{2}^{2}})\]
\[y = d_{2} = (\sqrt{2a^{2} + 2b^{2} - d_{1}^{2}})\]
Diagonal of a Cube Formula
For a cube, we find the diagonal by using a three-dimensional version of the Pythagorean Theorem/distance formula:
\[\text{The formula of diagonal of cube} = s\sqrt{3}\]
Where s is the side of a cube.
Diagonal of Cuboid Formula
\[\text{Diagonal of Cuboid Formula} = \sqrt{(l^{2} + b^{2} + h^{2})}\]
Where l, b, and h are length, breadth, and height of a cuboid
Solved Examples
Example 1: Find the diagonal of a rectangle with length as 8 cm and breadth value as 6 cm.
Solution: As given,
l = 8 cm
b= 6 cm
Formula for diagonal of rectangle is,
\[d =\sqrt{l^{2} + b^{2}}\]
Substituting the values,
\[d =\sqrt{8^{2} + 6^{2}}\]
\[d =\sqrt{(64 + 36)}\]
\[d = \sqrt{100}\]
d = 10 cm
Therefore, diagonal will be 10 cm.
Example 2: Find the length of the diagonal of a cube with side length as 10 cm?
Solution: As given,
a = 10 cm
The formula for diagonal of a cube is,
\[d = a\sqrt{3}\]
substituting the values,
\[d = 10 \times \sqrt{3}\]
i.e. d = 10 X 1.73
i.e.d = 17.3 cm
Therefore, diagonal will be 17.3 cm.
Historical Context
The word diagonal derives its meaning from the ancient Greek word διαγώνιος diagonios, meaning "from angle to angle". In the developmental stages of geometry, it was used by both Strabo and Euclid to refer to a line connecting two vertices of a polygon (such as a rhombus or cuboid). The word was later adopted into the Latin language as diagonus ( meaning "slanting line").
Non-Mathematical Uses of the Concept of Diagonal
This concept finds ample usage in the engineering background as a diagonal brace is used as a beam. These braces of a rectangular (cuboidal) structure (such as scaffolding) facilitates it to withstand strong forces pushing into it.
The understanding of diagonal is also useful practically in diagonal pliers which are devices that are wire-cutting. They are identified by the cutting edges of the jaws. These edges intersect at the joint rivet at an angle or "on a diagonal", hence the name.
A diagonal lashing is also a type of device used for lashing which helps to bind spars or poles.
In association with the sports football, the diagonal system of control is used by the match referees and assistant referees to appropriately position themselves in one of the four quadrants of the pitch.
Did you Know?
Polygons are quite ancient mathematical concepts that have been known since centuries. The regular polygons were known to the ancient Greeks, and evidence of their knowledge can be found in ancient structures that were built or mathematical models (statutes) preserved in museums.
In computer graphics, a polygon is said to be a primitive concept that is employed in modeling and rendering. They are more defined in a database system which consists of arrays of vertices (the coordinates of the geometrical vertices, as well as other attributes of the polygon.
The imaging system summons the structure of the polygon that is required for the particular scene, from the database. This is then transported in the active memory and finally, to the display system (such as a screen, TV monitors and the like) so that the scene is visualized. Note here that although polygons are two-dimensional structures, with the help of the imaging system in the computer, they are displayed in a visual scene in the correct three-dimensional orientation.
Quiz Time
A rectangle has a length of 14 cm and a width of 10cm. What is the diagonal of the rectangle?
How many diagonals does an eighteen-sided polygon have?
FAQs on Diagonal Formula
1. State Diagonals Examples in Real Life.
Some of the real-life examples of diagonals are:
Diagonals in squares and rectangles add strength to construction, whether for a house wall, bridge, or tall building.
You may see diagonal wires used to keep bridges steady.
When houses are being built, look for diagonal braces that hold the walls straight and true.
Bookshelves and scaffolding are braced with diagonals.
For a catcher in softball or baseball to throw out a runner at second base, the catcher throws along a diagonal from home plate to second.
The phone or computer screen is measured along its diagonal.
2. What is the Formula to Find the Number of Diagonals in any Polygon?
To find the number of diagonals in a polygon, we can use the formula below:
No. of diagonals in the polygon= n(n-3)/2
Where n= number of vertices (corners) in the polygon
Using this formula, we know that a triangle has zero diagonals, a quadrilateral has 2 diagonals, a pentagon has five diagonals and so on.
3. How can we use the diagonal formula? Mention the diagonal formula for a rhombus.
The diagonal formula is used for a polygon as stated below:
In the first step, we identify the polygon; whether it is a triangle, or a pentagon or a hexagon, etc.
In the next step, we evaluate the number of sides, vertices and other physical properties of the polygon.
Lastly, we apply the diagonal formula according to the evaluation done above. For example, for a square, we use the formula a√2 to calculate the length of the diagonal.
4. What is the diagonal formula of a square inscribed in a circle? Does a circle have diagonals?
For a square inscribed in a circle, the diagonal formula is simply what we would use for a normal square, that is, a√2, however, it has a special hint too. Usually for such a square the diagonal is equivalent to the diameter of the circle. Yes, a circle has infinitely many diagonals because it has infinite numbers of diameters which can join chords and vertices to form infinitely many polygons within the circle.
5. How do we define a diagonal for a polygon?
A diagonal may be defined as a straight line that when drawn connects to the two opposite ends of the polygon (known as vertex). A diagonal is possible for all polygonal structures. It is referred to as a line segment in geometry. It will be interesting to note that even a circle (which has no sides or corners) is said to possess diagonals. In fact, a circle is said to have infinitely many diagonals.
