Rhombus vs Parallelogram

Rhombuses and parallelograms both belong to the family of quadrilaterals. All types of quadrilaterals have some properties that bring them under the same category. However, each member of the family is different from the other in some aspects. Here we will discuss the differences between a rhombus and a parallelogram.

Let us compare the figure of a rhombus and a parallelogram below.

What is the Difference between a Rhombus and a Parallelogram

Basis	Rhombus	Parallelogram
SIDES	All sides are congruent	Opposite sides are equal, adjacent sides are unequal
DIAGONALS	Bisect each other at 90° and are equal	Bisect each other but not at 90° and are unequal
PERIMETER	It is given by P = 4s, where ‘s’ is any side	It is given by 2(a + b), where a and b are the side and the base respectively
AREA	It is given by  (d1 × d2)/2 where d1 and d2 are the length of the two diagonals	It is given by b × h where b and h represent the base and the height respectively
RELATION	A rhombus is a parallelogram since it satisfies all the properties a parallelogram exhibits	A parallelogram is not a rhombus as it does not satisfy the congruency of all its sides which a rhombus does

Similarities Between a Rhombus and a Parallelogram

• Have four sides
• Opposite sides equal and parallel
• Opposite angles equal
• Diagonals bisecting each other

Is a Rhombus a Parallelogram

YES! Remember, a parallelogram has 2 pairs of sides that are mutually opposite, parallel, and equal, and 2 diagonals mutually bisecting. Doesn’t a rhombus also satisfy this definition? YES. A rhombus satisfies all the properties of a parallelogram. So a rhombus is a parallelogram. Therefore, we can consider a rhombus as a subset of a parallelogram.

However, is a rhombus always a parallelogram?

YES! A rhombus is always a parallelogram.

A rhombus exhibits all the properties of a parallelogram. Also, it has all its sides congruent. Its all-sides congruency makes the diagonals bisect perpendicularly. Thus, we often consider a rhombus as a slanting square and a parallelogram as a slanting rectangle.

How to Prove a Parallelogram is a Rhombus

To prove: ABCD is a rhombus

Proof:

Given: ABCD is a parallelogram

AD ≅ BC

∠DAC ≅ ∠BCA

∠DEA = 90°

Steps	Statement	Reason
1	∠AED ≅ ∠BEC	Vertical angles are ≅
2	Δ AED ≅ ΔBEC	Angle – Angle Side (AAS) Postulate
3	AE ≅ EC	Corresponding Parts of Congruent Triangles (CPCTC) postulate
4	DE ≅ EB	CPCTC postulate
5	AC ⊥ Bis BD BD ⊥ Bis AC	Definition of perpendicular bisector
6	ABCD is a rhombus	Definition of rhombus Hence Proved

Now think the other way round. Are parallelograms rhombuses?

NO. A parallelogram does not have all its sides congruent, unlike a rhombus.