Formula for calculating the diagonal of a rhombus

A rhombus is a quadrilateral with four equal sides, a quadrilateral with two diagonals perpendicular to each other at the midpoint of each line is a rhombus, a parallelogram with two adjacent sides equal... Besides squares, rectangles, triangles... the rhombus is one of the important shapes in mathematics and life.

Besides the formula for calculating the perimeter and area of ​​a rhombus , the way to calculate the diagonal of a rhombus - the line connecting the opposite vertices of the rhombus together, is also very important.

The article below will help you learn how to calculate the diagonal of a rhombus with specific examples, please refer to it.

Table of Contents

• Diagonal of a rhombus
• Properties of the diagonals of a rhombus
• Formula for calculating the diagonal of a rhombus
• Problem on calculating diagonal of rhombus

Diagonal of a rhombus

• The diagonal of a rhombus is the line connecting two opposite vertices of the rhombus.
• A rhombus has two diagonals, and they intersect at the midpoint of the rhombus.
• The diagonal divides the rhombus into two equilateral triangles with equal sides.

Properties of the diagonals of a rhombus

The two diagonals in a rhombus have the following properties:

• Two equal diagonals: The two diagonals of a rhombus are of equal length.
• The angle between two diagonals is a right angle: The two diagonals of a rhombus intersect at the midpoint of the rhombus and form a right angle.
• Diagonals are the axes of symmetry of a rhombus: Each diagonal of a rhombus is an axis of symmetry of the rhombus, dividing the rhombus into two symmetrical halves.
• Diagonals are diagonals of two equilateral triangles: Each diagonal of a rhombus is a diagonal of two equilateral triangles, formed by equal sides.
• The product of the lengths of the two diagonals is equal to the product of the lengths of the two sides of a rhombus: The product of the lengths of the two diagonals of a rhombus is equal to the product of the lengths of the two sides of a rhombus. That is, if we denote the diagonal by d and the sides by a and b, we have d² = a² + b².

These properties are characteristic properties of rhombuses and are used in many geometric problems involving rhombuses.

Formula for calculating the diagonal of a rhombus

Please consider the example below to derive the formula for calculating the diagonal of a rhombus.

Suppose we need to calculate the length of the diagonal of a rhombus ABCD with side a and an angle ABC = 60 degrees -> what is the formula for calculating the diagonal of a rhombus in this case?

Solution:

Because ABCD is a rhombus, all sides are equal to a.

Consider triangle ABC with: AB = BC = a

Again: ABC = 60 degrees => Triangle ABC is an equilateral triangle with side a.

=> AB = AC = BC = a

=> The length of the diagonal of the rhombus is AC = BD = a.

The above solution is one of the simplest and easiest to understand formulas for calculating the diagonal of a rhombus.

Formula for calculating the diagonal of a rhombus when knowing the area and the remaining diagonal

From the formula for calculating the area of ​​a rhombus:

S = (axb) : 2

We have the formula for diagonal length as follows:

a = S x 2 : b

or

b = S x 2 : a

In there:

• S is the area
• a and b are the lengths of the two diagonals

Use the geometric properties of a rhombus to calculate the length of the diagonal without using Pythagoras' theorem. Specifically:

The diagonal of a rhombus is the average of the two heights.

Diagonal = square root of (long height + short height)²

The diagonal of a rhombus is half the perimeter of the rhombus.

Diagonal = 1/2 x perimeter of rhombus.

Problem on calculating diagonal of rhombus

Problem 1: Given a rhombus with an area of ​​360 square centimeters, the length of one diagonal is 24 centimeters. Calculate the length of the second diagonal.

Solution:

According to the formula for the area of ​​a rhombus: axb : 2

We have the second diagonal: 360 x 2 : 24 = 30cm

Answer: 30cm

Problem 2:

A rhombus has an area of ​​4dm, the length of one diagonal is 3/5 dm. Calculate the length of the second diagonal. Solution:

The length of the second diagonal is:

(4 x 2) : 3/5 =40/3 (dm)

Lesson 3: The two diagonals of a rhombus have lengths of 160cm and 120cm. Calculate the height of the rhombus, knowing that the ratio between the height and the length of the rhombus's side is 24:25.

Solution:

The area of ​​the rhombus is: 160.120:2 = 9 600 (cm2).

Since the ratio between the height and the length of the side of a rhombus is 24:25, we can consider the height of the rhombus to be 24a and the side of the rhombus to be 25a.

Then we have the area of ​​the rhombus: 25a.24a = 9 600 a2 = 16 a = 4 cm.

The height of the rhombus is: 24.4 = 96 (cm).

So the height of the rhombus is 96cm.

Lesson 4: 

Given rhombus ABCD with side length 12.5cm, height 6.72cm and AC is smaller than BD. What are the lengths of diagonals AC and BD respectively?

Prize:

Apply the formula to calculate the area of ​​a rhombus: S = ha = 6.72 x 12.5 = 84cm.
=> 1/2 AC x BD = 84 => 2AC.BD = 336

Let O be the intersection of the two diagonals of a rhombus.

We have AOB is a right triangle at O ​​so AB2 = OA2 + OB2

In which, OA = 1/2 AC, OB = 1/2 BD

=> 12.52 = 1/4 (AC2 + BD2) <=> 625 = AC2 + BD2

AC2 + BD2 = 625 <=> AC2 + BD2+ 2AC.BD = 625 + 336 <=> (AC + BD)2 = 961 <=> AC + BD = 31 (1)

AC2 + BD2 = 625 <=> AC2 + BD2- AC.BD = 625 -336 <=> (BD - AC)2 = 289 <=> BD - AC = 17 (According to the problem BD > AC) (2)

From (1) and (2), we have:

BD = 24, AC = 7cm.

Lesson 5: 

Rhombus ABCD has sides equal to 10 units. Calculate the length of the diagonal of the rhombus.

Solution: The length of the diagonal of rhombus ABCD is:

Diagonal = square root of 2(10²) = square root of 200 = 14.14 units of length.

So the length of the diagonal of rhombus ABCD is 14.14 units of length.

Lesson 6:

Rhombus ABCD has a diagonal of 12 units. Calculate the perimeter of the rhombus.

Solution: Since a rhombus has four equal sides, its perimeter will be the sum of the lengths of the four sides, that is:

Perimeter = 4 x side length = 4 x 6 = 24 units of length.

So the perimeter of rhombus ABCD is 24 units of length.