What is the Formula for the Diagonal of a Rectangle?
A line segment that joins the two opposite vertices of a rectangle is known as a diagonal. A rectangle is divided into two identical right triangles by either of its diagonals. The diagonals of a rectangle are equal in length, and bisect each other. The formula to find the diagonal of a rectangle is,
\(d=\sqrt{w^2+l^2}\)
Here, the term ‘\(w\)’ refers to width, ‘\(l\)’ refers to length and ‘\(d\)’ refers to the length of the diagonal.
What is the Formula for the Diagonal of a Square?
The diagonals of a square are equal in length and bisect each other at right angles. By using a specific formula, we can find the length of the diagonal of a square.
\(d=\sqrt2a\)
Where, ‘\(a\)’ is the side length and ‘\(d\)’ is the length of the diagonal.
Rapid Recall
Solved Examples
Example 1:
What is the length of the diagonal of a rectangle, if its length is 15 cm and its width is 18 cm?
Solution:
As stated, the given rectangle has the length,
\(l=15\) cm and width, \(w\) = 18 cm
The diagonal length of the rectangle is determined by using the formula,
\(d=\sqrt{w^2+l^2}\)
\(=\sqrt{{(18)}^2+{(15)}^2}\)
\(=\sqrt{324+225}\)
\(=\sqrt{549}\)
\(=23.43\)
Therefore, the length of the diagonal of a rectangle is 23.43 cm.
Example 2:
John has ordered a cake from a bakery nearby. All the four sides of the cake box are equal and have a length of 30 cm. What would be the diagonal length of the cake box?
Solution:
As mentioned the side length of the cake box, a = 30 cm
It is given that the cake box has four equal sides, which means it is a square. By using the formula to find the length of the diagonal of a square, we can find diagonal length with,
\(d=\sqrt2a\)
= \(\sqrt2\times30\)
= \(1.414\times30\) [Substitute the value of \(\sqrt2\)]
= 42.42
Thus, the length of the diagonal of the square box is 42.42 cm.
Example 3:
Rosy purchased some coloured papers to decorate her room. In order to make some paper craft, she cut the sheets of paper into squares and folded it such that the diagonals measured 45 cm. Can you find the side length of the square sheets of paper?
Solution:
It is stated that Rosy has cut the sheets of paper in the shapes of squares, which means that all the sides of the square pieces are equal.
Since the length of the diagonal is 45 cm, that is, d = 45 cm.
By using the formula to find the diagonal of a square, we can find the side lengths of the square pieces of paper.
So,
\(d=\sqrt2a\)
45 = \(\sqrt2a\)
\(\frac{45}{\sqrt2}=a\) [Divide both sides by \(\sqrt2\)]
\(\frac{45}{1.414}=a\) [Substitute the value of \(\sqrt2\)]
31.82 = \(a\)
Therefore, the colored square pieces of paper have side lengths of 31.82 cm.
A major difference between a rectangle and a square is that a square has sides of equal lengths, but a rectangle only has opposite sides that are equal.
Yes, the diagonals of a rectangle bisect each other and have equal lengths.
Yes, a square is a unique kind of rectangle. This is because it has all the characteristics of a rectangle, such as, having equal and parallel opposite sides with right angles as the interior angles.
A square has two diagonals that are equal in length. These diagonals intersect at the center of the square.