Difference Between a Square and a Rectangle - Properties Compared
Difference Between a Square and a Rectangle - Properties Compared
TL;DR
The core difference between a square and a rectangle is the sides: a square has all four sides equal, while a rectangle has only its opposite sides equal. Both have four right angles and equal diagonals, but a square's diagonals cross at 90° and a rectangle's do not. This guide compares every property with formulas and examples.
What is the difference between a square and a rectangle?
A square has four equal sides; a rectangle has two pairs of equal opposite sides but adjacent sides of different lengths. Both shapes are quadrilaterals, both have four right angles, and both have opposite sides parallel. The square simply adds one extra rule: all sides equal, not just opposite ones. That is why a square is best understood as a special rectangle: a rectangle that happens to have equal length and width.
Properties of A Square
A square is a quadrilateral with four equal sides and four right angles. Its properties are:
- All four sides are equal and opposite sides are parallel.
- All four angles are 90°.
- The diagonals are equal in length, bisect each other, and cross at 90°.
- Each diagonal bisects the corner angles into two 45° halves.
The side length s is the only measurement a square needs: area, perimeter, and diagonal all follow from it.
Properties of a Rectangle
A rectangle is a quadrilateral with opposite sides equal and four right angles. Its properties:
- Opposite sides are equal and parallel; adjacent sides are generally different (length l, width w).
- All four angles are 90°.
- The diagonals are equal in length and bisect each other, but they do not cross at 90° (they meet at an oblique angle unless the rectangle is a square).
- A rectangle needs two measurements, length and width, to be fully described.
You can read each shape on its own in the properties of a rectangle and the square in geometry.
Square vs Rectangle: The Comparison Table
| Property | Square | Rectangle |
|---|---|---|
| Sides | All four equal | Opposite sides equal |
| Angles | All 90° | All 90° |
| Measurements needed | One (side s) | Two (length l, width w) |
| Diagonals (length) | Equal | Equal |
| Diagonals (crossing angle) | 90° | Not 90° (oblique) |
| Area | s² | l×w |
| Perimeter | 4s | 2(l+w) |
| Diagonal length | s√2 | √(l² + w²) |
| Lines of symmetry | 4 | 2 |
Formulas For A Square And A Rectangle
Here is each formula with what its symbols mean. For a square, s is the side. For a rectangle, l is the length and w is the width.
Area: Square: A=s² Rectangle: A=l×w
Perimeter: Square: P=4s Rectangle: P=2(l+w)
Diagonal length: Square: d=s√2 Rectangle: d=√(l² + w²)
Why is a square called a rectangle?
A rectangle is defined as a quadrilateral with four right angles and opposite sides equal. A square meets that definition exactly: it has four right angles and opposite sides equal (its opposite sides are equal because all its sides are). So a square satisfies every condition for being a rectangle, plus the extra condition that adjacent sides are also equal.
The relationship runs one way only: every square is a rectangle, but not every rectangle is a square. A rectangle becomes a square the moment its length equals its width.
Examples of Difference Between a Square and a Rectangle
Example 1
Find the area and perimeter of a square with side 6 cm, and of a rectangle with length 8 cm and width 3 cm.
Square: A=s²=6²=36 cm², P=4s=4×6=24 cm
Rectangle: A=l×w=8×3=24 cm², P=2(l+w)=2(8+3)=22 cm
Final answer: Square: 36 cm², 24 cm. Rectangle: 24 cm², 22 cm.
Example 2
A shape has four right angles, and its opposite sides are equal. A student concludes it must be a square. Is that right?
Final answer: No. Those conditions describe a rectangle; it is a square only if all four sides are also equal.
Example 3
Find the diagonal of a square with side 5 cm.
Final answer: 5√2 cm, about 7.07 cm.
Example 4
Find the diagonal of a rectangle with length 12 cm and width 5 cm.
Final answer: 13 cm.
Example 5
A square and a rectangle have the same perimeter of 40 cm. The rectangle is 12 cm long. Which has the larger area?
Final answer: The square has the larger area (100 cm² vs 96 cm²).
Example 6
A rectangle has area 48 cm² and length 8 cm. Find its width, then state the condition under which it would be a square.
Final answer: Width = 6 cm; it would be a square only if length equalled width.
Where The Square-Versus-Rectangle Distinction Matters
The "for a fixed perimeter, the square holds the most area" result from Example 5 is why animal pens, storage tanks, and shipping crates trend toward square cross-sections when material (the perimeter) is the cost and capacity (the area) is the goal.
The diagonal difference matters in construction. A builder squaring up a foundation measures both diagonals: equal diagonals confirm true right angles.
The deeper point is that a more specific shape inherits every property of the general one and adds constraints. Knowing a square is a rectangle means every rectangle theorem is already proved for squares.
Tripping Points To Avoid
Mistake 1: Treating square and rectangle as mutually exclusive
The correct way: Every square is a rectangle; only some rectangles are squares.
Mistake 2: Confusing the diagonal rules
The correct way: Both shapes have equal diagonals that bisect each other. Only the square's diagonals cross at 90°.
Mistake 3: Using the wrong area formula
The correct way: A square needs only its side: A=s². A rectangle needs both length and width: A=l×w.
Conclusion
- The difference between a square and a rectangle is the sides: a square has all four equal, a rectangle only its opposite sides.
- Both have four right angles, parallel opposite sides, and equal diagonals.
- Only a square's diagonals cross at 90°; a rectangle's cross obliquely.
- Every square is a rectangle, but not every rectangle is a square.
- A square uses A=s² and a rectangle uses A=l×w; they match when length equals width.