Table of Contents

Last modified on August 3rd, 2023

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.

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 (d_{1} × d_{2})/2 where d_{1} and d_{2} 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 |

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

**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.

**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.

Last modified on August 3rd, 2023

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