Finding the perimeter of a rhombus: formula and tasks

In this publication, we will consider how to calculate the perimeter of a rhombus and analyze examples of solving problems.

Perimeter Formula

1. By the length of the side

The perimeter (P) of a rhombus is equal to the sum of the lengths of all its sides.

P = a + a + a + a

Because all sides of a given geometric figure are equal, the formula can be represented as follows (side multiplied by 4):

P = 4*a

2. By the length of the diagonals

The diagonals of any rhombus intersect at an angle of 90° and are divided in half at the point of intersection, i.e.:

• AO=OC=d₁/2
• BO=OF=d₂/2

The diagonals divide the rhombus into 4 equal right triangles: AOB, AOD, BOC and DOC. Let’s take a closer look at AOB.

You can find the side AB, which is both the hypotenuse of the rectangle and the side of the rhombus, using the Pythagorean theorem:

AB² = AO² + OB²

We substitute into this formula the lengths of the legs, expressed in terms of half the diagonals, and we get:

AB² = (d₁/ 2)² + (d₂/ 2)², or

So the perimeter is:

Examples of tasks

Task 1

Find the perimeter of a rhombus if its side length is 7 cm.

Decision:

We use the first formula, substituting a known value into it: P u4d 7 * 27 cm uXNUMXd XNUMX cm.

Task 2

The perimeter of the rhombus is 44 cm. Find the side of the figure.

Decision:

As we know, P = 4*a. Therefore, to find one side (a), you need to divide the perimeter by four: a = P / 4 = 44 cm / 4 = 11 cm.

Task 3

Find the perimeter of a rhombus if its diagonals are known: 6 and 8 cm.

Decision:

Using the formula in which the lengths of the diagonals are involved, we get: