Rhombus perimeter calculator

What is a rhombus and its perimeter?

A rhombus is a quadrilateral where all sides are equal in length. It is a special type of parallelogram where opposite sides are parallel, and the angles can differ, unlike in a square where all angles are 90 degrees. The perimeter of a rhombus is the sum of the lengths of all its sides. To calculate the perimeter of a rhombus, you need to know the length of at least one side or the lengths of the diagonals.

Importance of understanding rhombus perimeter

Understanding how to calculate the perimeter of a rhombus can be useful in a variety of situations. In geometry and construction, knowledge about the properties of a rhombus is necessary for design and material calculations. It is also important for students studying geometry, as the rhombus is a classic example in problem tasks.

Formulas

Perimeter of a rhombus by side

The simplest way to calculate the perimeter of a rhombus is by using the length of one of its sides. The formula for the perimeter given the side length $a$ is:

$$P_{\text{rhombus}} = 4a$$

where $a$ is the length of the side of the rhombus.

Perimeter of a rhombus by diagonals

If the lengths of the diagonals $d_1$ and $d_2$ are known, the perimeter can be found using the following formula:

$$P_{\text{rhombus}} = 2 \times \sqrt{d_1^2 + d_2^2}$$

This formula is derived from applying geometric properties of the rhombus. First, the side length $a$ is determined using the Pythagorean theorem, as the diagonals intersect at right angles:

$$a = \sqrt{\left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2} = \frac{1}{2} \times \sqrt{d_1^2 + d_2^2}$$

Substituting this value into the perimeter formula gives:

$$P_{\text{rhombus}} = 4a = 2 \times \sqrt{d_1^2 + d_2^2}$$

Note that knowing the diagonals allows you to also compute the area of the rhombus using the rhombus area calculator.

Examples

Example 1: Calculate perimeter by side

Suppose the side length of the rhombus is 5 cm. Using the formula for the side, we obtain:

$$P_{\text{rhombus}} = 4 \times 5 = 20 \text{ cm}$$

Example 2: Calculate perimeter by diagonals

Suppose the diagonals are 6 cm and 8 cm. Using the formula for diagonals:

$$P_{\text{rhombus}} = 2 \times \sqrt{6^2 + 8^2} = 2 \times \sqrt{36 + 64} = 2 \times \sqrt{100} = 2 \times 10 = 20 \text{ cm}$$

Notes

• A rhombus is a parallelogram where all sides are equal, making the perimeter calculation simple when the side is known.
• The diagonals of a rhombus always intersect at right angles and divide it into four equal right triangles.
• Knowing any side or the diagonals allows you to quickly and easily calculate the perimeter of a rhombus.

Frequently asked questions

How to find the perimeter of a rhombus if only the area and one diagonal are known?

To find the perimeter, start by determining the length of the unknown diagonal. If the area $S$ and one diagonal $d_1$ are known, the other diagonal $d_2$ is determined using the formula for the area of a rhombus:

$$S = \frac{d_1 \cdot d_2}{2}$$

Thus,

$$d_2 = \frac{2S}{d_1}$$

After that, you can use the formula for the perimeter using the diagonals.

What is the perimeter of a rhombus with a side of 3.2 m?

If the side of the rhombus is 3.2 m, the perimeter can be calculated as:

$$P_{\text{rhombus}} = 4 \times 3.2 = 12.8 \text{ m}$$

Are all angles in a rhombus equal?

No, not necessarily. A rhombus can have two pairs of equal angles, but they are not required to be right angles, as in a square.

Can every rhombus be considered a square?

No, although all sides of a rhombus are equal, its angles do not have to be right. Only in a square, all angles are 90 degrees.

How to distinguish a rhombus from other types of parallelograms?

The main difference of a rhombus from other parallelograms is the equality of all its sides. In a regular parallelogram, opposite sides are equal, but not necessarily all four.

Is it useful to know the diagonals of a rhombus?

Yes, knowing the diagonals is useful for calculating the area and perimeter of a rhombus when the length of the sides is unknown.