Square and Rhombus: Properties, Shapes, and Key Comparison
Understanding the difference between square and rhombus is important in academic geometry, as both are fundamental quadrilaterals but differ in distinct mathematical properties. This comparison clarifies their definitions, structure, and usage in mathematics, which is essential for students in Classes 8–12 and for JEE exam preparations.
Meaning of Square in Mathematics
A square is a special type of quadrilateral where all four sides are equal and every angle measures exactly 90 degrees. It is a regular polygon with high symmetry and specific properties linked to its angles and sides.
The diagonals of a square are equal, bisect each other at right angles, and each diagonal divides the square into two congruent isosceles right triangles. For more insights, refer to Area Of Square Formula.
$Area = s^2$
Understanding the Rhombus
A rhombus is a parallelogram in which all four sides are of equal length, but the angles are not necessarily right angles. It is often called a diamond in everyday language due to its shape.
The diagonals of a rhombus are unequal but bisect each other perpendicularly. Opposite angles are equal, and adjacent angles are supplementary.
$Area = \dfrac{1}{2} d_1 d_2$
Comparative View of Square and Rhombus
| Square | Rhombus |
|---|---|
| All four angles are 90 degrees | Angles are not all 90 degrees |
| All sides are equal in length | All sides are equal in length |
| Diagonals are equal | Diagonals are unequal |
| Diagonals bisect at right angles | Diagonals bisect at right angles |
| Each diagonal divides the square into two congruent triangles | Each diagonal divides rhombus into two congruent triangles |
| Has four lines of symmetry | Has two lines of symmetry |
| Can be inscribed and circumscribed about a circle | Cannot always be inscribed or circumscribed about a circle |
| Area formula: side × side | Area formula: ½ × product of diagonals |
| All internal angles are congruent | Only opposite angles are congruent |
| All sides are both congruent and perpendicular | Sides are congruent but not perpendicular |
| Square is a type of rhombus | Every rhombus is not a square |
| Diagonals are axes of symmetry | Diagonals are axes of symmetry |
| Perimeter is 4 × side length | Perimeter is 4 × side length |
| Can be constructed inside a regular circle | Cannot always be constructed in a regular circle |
| Has rotational symmetry of 90°, 180°, 270°, 360° | Has rotational symmetry of 180°, 360° only |
| Opposite sides are parallel and equal | Opposite sides are parallel and equal |
| All four angles are equal | Opposite angles are equal |
| Length of diagonals: $s\sqrt{2}$ | Length of diagonals differ and use varied formulas |
| Sum of interior angles: 360° | Sum of interior angles: 360° |
| Greater symmetry than rhombus | Less symmetry than square |
Core Distinctions
- All square angles are right angles; rhombus angles vary
- Square diagonals are equal; rhombus diagonals are unequal
- Square has higher symmetry than rhombus
- Not every rhombus is a square
- Square sides are perpendicular; rhombus sides are not
Simple Numerical Examples
A square with side 6 cm has area $6^2 = 36$ cm², and diagonal $6\sqrt{2}$ cm. This demonstrates square properties discussed earlier and for deeper insights refer to Difference Between Square And Rectangle.
A rhombus with diagonals 10 cm and 8 cm has area $\dfrac{1}{2} \times 10 \times 8 = 40$ cm², with unequal diagonals and opposite angles equal.
Applications in Mathematics
- Used in geometric constructions and symmetry analysis
- Important in problems involving tessellation of the plane
- Appear in quadratic equations and coordinate geometry
- Useful in mensuration for calculating area and perimeter
- Serve in design and architecture for tile patterns
Summary in One Line
In simple words, a square has equal sides and all right angles, whereas a rhombus has equal sides but does not necessarily have right angles.
FAQs on What Is the Difference Between a Square and a Rhombus?
1. What is the main difference between a square and a rhombus?
The main difference between a square and a rhombus is that all angles in a square are right angles, while in a rhombus, only sides are equal but angles are not necessarily right angles.
- Square: Four equal sides and four right angles (each 90°)
- Rhombus: Four equal sides but opposite angles are equal, not necessarily 90°
- Both are parallelograms with perpendicular diagonals
2. List the similarities between a square and a rhombus.
A square and a rhombus share many properties since both are types of parallelograms.
- All four sides are of equal length in both shapes
- Opposite sides are parallel
- Diagonals bisect each other at 90 degrees
- Diagonals bisect the angles
3. Is every square a rhombus? Explain.
Yes, every square is a rhombus because it satisfies all the properties of a rhombus.
- All four sides are equal
- Diagonals bisect each other at right angles
- Opposite sides are parallel
- But a square also has four right angles, which is not required for a rhombus
4. Can a rhombus be a square? Why or why not?
A rhombus can be a square if all its angles are right angles.
- All squares are rhombuses with all angles 90°
- A rhombus becomes a square only if every angle measures 90°
5. What are the angle properties of a square versus a rhombus?
Squares always have four right angles, while rhombuses have opposite angles equal but not necessarily 90°.
- Square: All angles are exactly 90°
- Rhombus: Opposite angles are equal, adjacent angles are supplementary
6. How do the diagonals of a square and a rhombus differ?
Both square and rhombus have diagonals that bisect each other at right angles, but only in a square are the diagonals equal in length.
- Square: Diagonals are equal and perpendicular
- Rhombus: Diagonals are perpendicular but generally of different lengths
7. What are the formulas for the area of a square and a rhombus?
The area formulas for square and rhombus are based on their side lengths and diagonals.
- Square: Area = side × side (s2)
- Rhombus: Area = (1/2) × diagonal1 × diagonal2
8. Name some real-life examples of squares and rhombuses.
Squares and rhombuses appear in daily life in various forms.
- Square examples: Chessboard squares, window panes, floor tiles
- Rhombus examples: Kite shapes, diamond signs on roads, certain quilt patterns
9. Why is a square called a regular quadrilateral but a rhombus is not?
A square is called a regular quadrilateral because all sides and all angles are equal; a rhombus only has equal sides.
- Square: All angles = 90°, all sides equal (regular)
- Rhombus: All sides equal, angles not necessarily 90° (not regular)
10. State any two differences between a square and a rhombus.
Two important differences are:
- In a square, each angle is 90°, while in a rhombus angles can be different.
- The diagonals of a square are equal, but in a rhombus they are usually unequal.
