Formula for calculating the diagonal of a square and the diagonal of a rectangle

How to calculate the diagonal of a square, how to calculate the diagonal of a rectangle is used a lot in math problems and practical applications such as design and construction, cutting corners, measuring... Quantrimang.com has compiled knowledge about the properties of two diagonals as well as calculation formulas, please refer to apply in study, life and work.

Table of Contents

• What is the diagonal of a square and a rectangle?
• How to calculate the diagonal of a square
  • Properties of square diagonal
  • Formula for calculating the diagonal of a square
  • Example of calculating the diagonal of a square 
• How to calculate the diagonal of a rectangle
  • Properties of diagonals of a rectangle
  • Formula for calculating the diagonal of a rectangle
  • Example of calculating diagonal of rectangle 
  • Example to prove that a quadrilateral is a rectangle.

What is the diagonal of a square and rectangle?

The diagonal of a square or rectangle is a straight line connecting two opposite corners. Each square or rectangle has two diagonals of equal length.

How to calculate the diagonal of a square

Properties of square diagonal

• The two diagonals of a square are equal in length, perpendicular, and intersect at the midpoint of each.
• There is an inscribed circle and a circumscribed circle and the centers of both circles coincide and are the intersection of the two diagonals of the square.
• 1 diagonal will divide the square into two right isosceles triangles.
• The intersection of the angle bisectors, medians, and perpendicular bisectors all coincide at one point.
• Has all the properties of a rectangle, parallelogram and rhombus.

Formula for calculating the diagonal of a square

According to the properties of a square, two diagonals of a square are equal and one diagonal of a square will divide the square into two parts of equal area, which are two isosceles right triangles. Thus, the diagonal of the square is the hypotenuse of the two isosceles right triangles.

So to calculate the diagonal of a square, you just need to apply the Pythagorean theorem to the right triangle.

Suppose you have square ABCD with side length a, diagonal AC divides the square into two right-angled triangles ABC and ACD.

Applying the Pythagorean theorem to the right isosceles triangle ABC:

⇒ ⇒

So the diagonal of a square has side length a:

Example of calculating the diagonal of a square 

Example 1:  A square has a side of 3cm. Is the diagonal of the square: 6cm, √18cm, 5cm, or 4cm?

Solution:

a) Applying the Pythagorean theorem in square ABC, we have:

AC² = AB² + BC² = 3² + 3² = 18

=> AC = cm

So the diagonal of the square is √18 cm.

Example 2:

The diagonal of a square is 2dm. The side of the square is: 1cm, 3/2cm, √2cm or 4/3cm?

Prize:

Apply the Pythagorean theorem to the right triangle ABC, but this exercise gives the length of the diagonal, i.e. AC = 2cm, calculate side AB.

We have: AC² = AB² + BC² = 2AB (because AB = BC)

=> AB² = AC²/2 = 2²/2 = 2

=> AB = √2

How to calculate the diagonal of a rectangle

A rectangle is a convex quadrilateral with four right angles, it is a parallelogram with two equal diagonals.

Properties of diagonals of a rectangle

The diagonals of a rectangle have several important properties that are useful in solving problems involving rectangles and their diagonals.

• The length of the diagonal of a rectangle is the hypotenuse of a right triangle, so it is equal to the square root of the sum of the squares of the two sides.
• The diagonal divides the rectangle into two right triangles of equal area. Therefore, the diagonal of a rectangle is the axis of symmetry of the rectangle.
• The two diagonals of a rectangle are equal and intersect at the midpoint of each line and form 4 isosceles triangles.

Formula for calculating the diagonal of a rectangle

From the properties of the diagonal of a rectangle above, we can use the Pythagorean theorem to calculate the length of the diagonal of a rectangle.

Suppose you have a rectangle ABCD with length a and width b, diagonal AC as shown below.

We apply the Pythagorean theorem to right triangle ABC:

⇒ ⇒

Therefore, the diagonal of a rectangle with length a and width b is:

So, the length of a diagonal of a rectangle is equal to the square root of the sum of the squares of the two sides (length and width) of the rectangle.

Thus, we can calculate the diagonal of a square or rectangle by simply applying the Pythagorean theorem.

Example of calculating diagonal of rectangle 

Calculate the length of the diagonal of a rectangle with a length of 10dm and a width of 5dm.

Solution:

Let the length of the diagonal of the rectangle be a (a > 0, dm)

Applying the Pythagorean theorem, the length of the diagonal of the rectangle is:

a2 = 102 + 52 = 125

=> a = 5√5 dm

Example to prove that a quadrilateral is a rectangle.

The properties and formula for calculating the diagonal of a rectangle can be applied to solve some problems proving that a quadrilateral is a rectangle.

Given quadrilateral ABCD with two diagonals perpendicular to each other. Let E, F, G, H be the midpoints of sides AB, BC, CD, AD respectively. What kind of shape is quadrilateral EFGH? Why?

Solution:

Because E is the midpoint of AB, H is the midpoint of AD

=> EH is the median of triangle ABD.

(1)

Because F is the midpoint of BC, G is the midpoint of CD

=> FG is the median of triangle BCD

(2)

From (1) and (2) =>

Consider the quadrilateral EFGH.

FG // EH

FG = EH

=> EFGH is a parallelogram (recognition sign)

On the other hand:

Also has:

E is the midpoint of AB, F is the midpoint of BC.

=> EF is the median of triangle ABC

=> EF // AC

But EH ⊥ AC => EH ⊥ EF

Parallelogram EFGH has a right angle

=> EFGH is a rectangle