Diagonal of Rectangle
The diagonal of rectangle is a line segment that connects the two opposite vertices of the rectangle. The diagonal of the rectangle divides the rectangle into two equal right triangles. A rectangle is a twodimensional figure with four sides, four vertices, and four angles. The two diagonals of a rectangle have the same length.
In this article, we will understand the concept of diagonal of a rectangle and the method to calculate its length. We shall explore the properties, formula, and derivation of the formula to evaluate the length of the diagonal of a rectangle. Let us learn more about the diagonals of a rectangle, their formula, and solve a few examples.
What is Diagonal of Rectangle?
The diagonal of a rectangle is a line segment that joins any two of its nonadjacent vertices. A rectangle has two diagonals where each of the diagonals divides the rectangle into two rightangled triangles with the diagonal being the hypotenuse. The diagonals bisect each other, making one obtuse angle and the other an acute angle. The two triangles formed by the diagonal of rectangle are congruent.
Diagonals of Rectangle Properties
The diagonal of rectangle is a line segment drawn between the opposite vertices of the rectangle. The properties of diagonals of a rectangle are as follows:
 The two diagonals of a rectangle are congruent. In other words, the length of the diagonals is equal.
 The two diagonals bisect each other and divide the rectangle into two equal parts.
 The length of the diagonal of rectangle can be obtained using the Pythagoras theorem.
 When the diagonals bisect each other, the angles of a rectangle at the center become one obtuse angle and the other an acute angle.
 When two diagonals bisect each other at 90° it is called a square.
 Since the diagonal of rectangle divide the rectangle into two rightangled triangles, it is considered the hypotenuse of these triangles.
Diagonal of Rectangle Formula
The diagonal of a rectangle formula helps in finding the length and width of the rectangle. In the following rectangle, AC and BD are the diagonals. You can see that the lengths of both AC and BD are the same. A diagonal cuts a rectangle into 2 right triangles, in which the sides are equal to the sides of the rectangle and with a hypotenuse. That hypotenuse is the diagonal.
Thus, to calculate the length of diagonals of a rectangle the formula is: d = √( l^{2} + w^{2})
Where,
 d = diagonal of rectangle
 l = length of the rectangle
 w = width of the rectangle
Diagonal of Rectangle Derivation
The diagonal of a rectangle formula is derived using the Pythagoras theorem. Consider the rectangle given above. Let us consider a rectangle of length "l" and width "w". Let the length of each diagonal be "d".
Applying Pythagoras theorem to the triangle BCD,
d^{2} = l^{2} + w^{2}
Taking square root on both sides,
d = √( l^{2} + w^{2})
Thus, the diagonal of a rectangle formula is:
d = √( l^{2} + w^{2})
Where,
 l = length of the rectangle
 w = width of the rectangle
Angles of Diagonal of Rectangle
The diagonals of a rectangle are of equal length and they bisect each other but do not form right angles at the center. They form linear pairs of angles such as obtuse angle + acute angle at each of the diagonal. The rectangle is called a square if its diagonals bisect each other at right angles as the diagonals of a rectangle do not bisect the respective vertex angles into equal angles.
A diagonal divides a rectangle into two congruent triangles, that too right triangles with their hypotenuse being the same. Each diagonal acts as the hypotenuse for the right triangles so formed. Applying Pythagoras theorem to the triangle so formed, d^{2} = l^{2} + w^{2}, where d is diagonal, l is length and w is the width of the rectangle. Taking square root on both sides, √(d^{2}) = √( l^{2} + w^{2}). Thus, the diagonal of a rectangle formula is: √( l^{2} + w^{2}).
Important Notes on Diagonal of Rectangle
 The diagonal of rectangle is a line segment joining the nonadjacent vertices.
 We can calculate the length of the diagonal of rectangle using the formula, d = √( l^{2} + w^{2}).
 The two diagonals of rectangle bisect each other.
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Diagonals of Rectangle Examples

Example 1: Using the diagonal of rectangle formula, find the diagonal of a rectangle whose dimensions are 3 units and 4 units.
Solution:
The dimensions of a rectangle = 3 units and 4 units.
According to the properties of angles of the rectangle, the diagonal of a rectangle = (d) = √( l^{2} + w^{2})
Length of diagonal of rectangle = √(3)^{2}+(4)^{2} = √25 = 5 units.Answer: Therefore, the length of the diagonal of the rectangle is 5 units.

Example 2: The size of the screen of a television is the length of its diagonal. Then find the size of the television whose dimensions are 18 inches and 46 inches.
Solution:
The length of the television, l = 46 units.
The width of the television, w = 18 units.
Using the diagonal of a rectangle formula,
d = √( l^{2} + w^{2})
d = √( 46^{2} + 18^{2})
= √2440
= 49.39 inches.
Answer: Therefore, the size of the television is 49.39 inches.

Example 3: If a rectangular board has a length of 13 inches with a diagonal of 18 inches. What is the width of the rectangle?
Solution:
The length of the board, l = 13 inches.
The diagonal of the board, d = 18 inches.
Using the diagonal of rectangle formula,
d = √( l^{2} + w^{2})
18^{2} = (13^{2} + w^{2})
w = √155
w = 12.4 inches.
Answer: Therefore, the width of the rectangle is 12.4 inches.
FAQs on Diagonals of Rectangle
What is the Diagonal of Rectangle?
The diagonal of a rectangle is a line segment that is drawn from the opposite vertices of the rectangle and bisect the other diagonal. There are two diagonals of a rectangle that are of the same length and divide the rectangle into two equal parts. The diagonal of the rectangle divides the rectangle into two rightangled triangles with a hypotenuse.
What is the Diagonal Formula of Rectangle?
The formula to calculate the diagonal of rectangle helps in finding the length of the diagonal. The formula is:
d = √( l^{2} + w^{2})
Where,
 d = diagonal of rectangle
 l = length of the rectangle
 w = width of the rectangle
What are the Properties of Diagonal of Rectangle?
The properties of the diagonal of rectangle are:
 The length of the two diagonals is equal.
 The two diagonals bisect each other and divide the rectangle into two equal parts.
 The length of the diagonals can be obtained using the Pythagoras theorem.
 When the diagonals bisect each other, the angles at the center of a rectangle become one obtuse angle and the other an acute angle.
 When two diagonals bisect each other at 90° it is called a square.
Are Diagonals of Rectangle Equal?
The diagonals of a rectangle bisect each other and are equal. Since a rectangle is a quadrilateral, all the angles are right angles. A rectangle is also a parallelogram where the opposite sides are equal.
Do Diagonals of Rectangle Bisect at 90°?
The diagonals of a rectangle bisect each other but not at 90 degrees angle.
How to Determine the Area of Rectangle When Diagonal of Rectangle is Given?
We can calculate the area (A) of a rectangle when its diagonal (d) and length (l) is given using the formula, Area = l × √(d^{2}  l^{2}).
How to Find the Length of Rectangle When The Diagonal of Rectangle is Given?
We can calculate the length of rectangle (l) when its diagonal (d) and width (w) is given using the formula, l = √(d^{2}  w^{2}).
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