# What is the relation between the diagonals of a rhombus?

It is the subject of mathematics that deals with the study of geometrical shapes, calculations and figures. It discusses all equations and formulas based on different geometrical shapes and figures. It deals with both 2-dimensional shapes and 3 Dimensional figures. It deals with some of the parameters of geometrical shapes such as length, height, Area, width, lateral surface area, total surface area, etc.

### Rhombus

Rhombus is a 2 D quadrilateral. It is a special type of parallelogram whose all 4 sides are of equal length. It generally looks like a diamond hence also called a diamond.

- All rhombus is quadrilateral but all quadrilaterals are not rhombus.
- All rhombus is parallelogram but all parallelograms are not a rhombus.
- All squares are rhombus but all rhombus is not square.

**Properties of the rhombus**

- All sides are equal.
- Opposite sides are parallel to each other.
- Opposite angles are equal.
- Any two adjacent angles sum to 180.
- Diagonals bisect at 90.
- Diagonals bisect the angles.
- If the mid-points are joined of the sides of the rhombus in order we get a rectangle.
- The Sum of all interior angles of the rhombus is 360.
- The area of the rhombus is given by 1/2 × d
_{1}× d_{2}where d_{1 }and d_{2}are the Diagonals. - The perimeter of the rhombus is given by 4a where a is the side of the rhombus.

### What is the relation between the diagonals of a rhombus?

**Solution:**

Following are the relation between diagonals of the rhombus

- Area of the rhombus = 1/2 × (a × b) where a and b are the length of the diagonal.
- Diagonals bisect each other at right angles.
- The Diagonals bisect the angles.
- Each diagonal divides the rhombus into two congruent triangles.

### Sample Problems

**Question 1: Given the side of an equal-sided parallelogram is 5 cm, find the perimeter of the given shape.**

**Solution: **

Since it is a parallelogram whose all sides are equal hence it can be either square or a rhombus. In both cases, the perimeter is 4 times the sides.

Hence, the perimeter of the given figure is (4 × 5) = 20 cm.

**Question 2: The length of the two diagonal is 12 cm and 24 cm. Find the area of the rhombus.**

**Solution:**

Given:

Diagonal d

_{1}= 12 cm and Diagonal d_{2 }= 24 cmThe Area of the rhombus, if given as (d

_{1}× d_{2})/2 square unitsHence applying the formula,

Area = (24 × 12)/2

Area = 144 cm

^{2}Therefore, the area of the rhombus is 144 cm

^{2}

**Question 3: The perimeter of the rhombus is 64 cm. Find the sides of the rhombus.**

**Solution:**

The perimeter of the rhombus is given by 4 × (length of the side).

Let’s say the length is a cm.

4 × a = 64.

a = 16 cm

The length of the side of the rhombus is 16 cm.