Diagonal Formula

Diagonals in squares and rectangles are adding strengths to construction, whether for a house wall, bridge, or tall building. We can see diagonal wires used to keep bridges steady and stable. When houses are being built, we can look for diagonal braces that hold the walls straight and true. The phone or computer screen we are viewing is measured along its diagonal. For example, a 21-inch screen tells that it is 21 inches from one corner to an opposite corner. In this article, we will discuss the diagonals of various shapes and their diagonal formula. Let us begin it!

What’s a Polygon?

A polygon is any two-dimensional plane shape made only with straight sides that close in a space. Also, sides that do not cross each other.  A triangle is a polygon with minimum sides. A dart, kite, quadrilateral, and star are all some kind of polygons. Simple polygons can be in a concave or convex shape.

What is Diagonal?

A diagonal of a polygon is a straight line from one vertex to a non-adjacent vertex. Also, the triangle, the simplest polygon, has no diagonals. A quadrilateral, which has four sides is having two diagonals. Similarly, a pentagon, whether regular or irregular, has five diagonals.

In convex polygons, diagonals will always be within the interior. Let us consider a rectangular door. We can run a line from the top hinge corner to the bottom opposite corner. We can also run a line from the bottom hinge corner up to the top opposite corner. Those two will be diagonals possible.

In concave, simple polygons, the diagonals may go outside the polygon, crossing sides and partly lying in the shape’s exterior. They are still diagonals. Darts and stars are typical examples of concave polygons with diagonals outside their shapes.

Thus the line stretching from one corner of the square or rectangle to the opposite corner through the center point of the figure is known as the diagonal. Squares are having two diagonals and equal in length.

Diagonal Formula

(1) Diagonal Formula of a Rectangle

d = \(\sqrt{l^2+b^2}\)

d	Diagonal
l	Length
b	Breadth

(2) Diagonal formula of a Square

\(d= a\sqrt{2}\)

d	Diagonal
a	Length of one side

(3) Diagonal formula of a Cube

\(d= s\sqrt{3}\)

d	diagonal
s	Length of one side

(4) Diagonal formula of a Cuboid

\(d= \sqrt{l^2+b^2+h^2}\)

d	diagonal
l	Length
b	Breadth
h	Height

(5) Diagonal formula for a Parallelogram

If the length of one diagonal is known, then the formula for other diagonal is:

\(d_2=\sqrt{2a^2+2b^2-d_1^2}\)

d_1	First diagonal
d_2	Second diagonal
a	First side length
b	Second side length

Solved Examples

Q.1: You are required to find the diagonal of a rectangle with length value 8 cm and breadth value 6 cm.

Solution: As given,

l = 8 cm

b= 6 cm

Formula for diagonal of rectangle is,

\(d = \sqrt{l^2+b^2}\)

putting the values,

\(d = \sqrt{8^2+6^2}\)

i.e. d= 10

Therefore, diagonal will be 10 cm.

Example-2: What will the length of the diagonal of a cube with side length as 8 cm?

Solution: As given,

a = 8 cm

Formula for diagonal of cube is,

\(d = a\sqrt{3}\)

putting the values,

\(d = 8\sqrt{3}\)

i.e. d= 8 X 1.73

i.e.d= 13.84 cm

Therefore, diagonal will be 13.84 cm.