A quadrilateral is any flat shape with four straight sides; all interior angles sum to 360°. KS3 pupils must know six types: square, rectangle, rhombus, parallelogram, trapezium, and kite. Each has a distinct combination of parallel sides, equal sides, equal angles, diagonal properties, and lines of symmetry.
What are the properties of a square and a rectangle?
Square:
- All four sides equal.
- All four angles equal 90°.
- Two pairs of parallel sides.
- Diagonals are equal in length, bisect each other, and cross at 90°.
- Four lines of symmetry; rotational symmetry order 4.
Rectangle:
- Two pairs of equal opposite sides.
- All four angles equal 90°.
- Two pairs of parallel sides.
- Diagonals are equal in length and bisect each other, but do not cross at 90° (unless it is a square).
- Two lines of symmetry (through midpoints of opposite sides); rotational symmetry order 2.
A square is a special rectangle — it satisfies every rectangle property and adds the condition that all sides are equal.
What are the properties of a rhombus and a parallelogram?
Rhombus:
- All four sides equal.
- Opposite angles equal; adjacent angles are supplementary (add to 180°).
- Two pairs of parallel sides.
- Diagonals bisect each other at 90° but are not equal in length.
- Each diagonal bisects the angles at its endpoints.
- Two lines of symmetry (along the diagonals); rotational symmetry order 2.
Parallelogram:
- Two pairs of equal and parallel opposite sides.
- Opposite angles equal; co-interior angles (between the parallel sides) add to 180°.
- Diagonals bisect each other but are not equal in length and do not cross at 90°.
- No lines of symmetry; rotational symmetry order 2.
A rhombus is a special parallelogram with all sides equal — just as a square is a special rectangle.
What are the properties of a trapezium?
A trapezium has exactly one pair of parallel sides (called the parallel sides or bases) and one pair of non-parallel sides.
Key facts:
- Co-interior angles between the parallel sides sum to 180° (e.g. if one base angle is 70°, the adjacent one on the same side is 110°).
- In an isosceles trapezium: the two non-parallel sides are equal, base angles are equal, and the diagonals are equal in length. It has one line of symmetry.
- A general trapezium has no lines of symmetry and no equal sides.
What are the properties of a kite?
A kite has two pairs of adjacent (neighbouring) equal sides — not opposite sides. Imagine two isosceles triangles sharing a common base.
Properties:
- One pair of opposite angles equal (the angles between the unequal sides).
- Diagonals cross at 90°.
- The longer diagonal (the "main" diagonal) bisects the shorter diagonal.
- One line of symmetry (along the main diagonal).
- Rotational symmetry order 1.
How do the six types compare at a glance?
| Quadrilateral | Pairs of parallel sides | All sides equal? | All angles 90°? | Diagonals equal? | Diagonals at 90°? | Lines of symmetry |
|---|---|---|---|---|---|---|
| Square | 2 | ✓ | ✓ | ✓ | ✓ | 4 |
| Rectangle | 2 | ✗ | ✓ | ✓ | ✗ | 2 |
| Rhombus | 2 | ✓ | ✗ | ✗ | ✓ | 2 |
| Parallelogram | 2 | ✗ | ✗ | ✗ | ✗ | 0 |
| Trapezium | 1 | ✗ | ✗ | ✗ | ✗ | 0 (or 1 if isosceles) |
| Kite | 0 | ✗ | ✗ | ✗ | ✓ | 1 |
How do you identify a quadrilateral from its properties?
Work through these questions in order:
- Does it have two pairs of parallel sides? If yes → square, rectangle, rhombus, or parallelogram.
- Are all sides equal? → rhombus (or square if also 90°).
- Are all angles 90°? → rectangle (or square if all sides also equal).
- Does it have exactly one pair of parallel sides? → trapezium.
- Does it have two pairs of adjacent equal sides? → kite.
- None of the above → irregular quadrilateral.
What are the most common mistakes at KS3?
- Saying a parallelogram has equal diagonals. It does not — only rectangles, squares, and isosceles trapeziums have equal diagonals.
- Thinking a rhombus has four right angles. It only has 90° angles if it is a square.
- Confusing "opposite" and "adjacent" sides. In a kite the equal sides are adjacent (next to each other), not opposite.
- Missing the difference between a trapezium and a parallelogram. A trapezium has exactly one pair of parallel sides; a parallelogram has two.
Frequently asked questions
Is a square a rectangle?
Yes. A square satisfies every property of a rectangle (two pairs of equal opposite sides, all angles 90°, two pairs of parallel sides) and additionally has all four sides equal. In mathematics, a square is a special case of a rectangle, just as a rectangle is a special case of a parallelogram.
Do the diagonals of a kite bisect each other?
Only partially. The main (longer) diagonal of a kite bisects the shorter diagonal at 90°, but the shorter diagonal does not bisect the longer one — it meets it at an off-centre point. In a rhombus or parallelogram, both diagonals bisect each other.
What is the angle sum of any quadrilateral?
Any quadrilateral's interior angles add to 360°. You can see why by drawing a diagonal to split the quadrilateral into two triangles: each triangle has an angle sum of 180°, giving 2 × 180° = 360° for the quadrilateral.
How is a rhombus different from a square?
Both have four equal sides, but a square has four right angles (90°) whereas a rhombus has opposite angles equal without the 90° requirement. A square is a rhombus with right angles; a rhombus is not necessarily a square.
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