Properties of Quadrilaterals: KS3 Maths Complete Guide

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?

How do you identify a quadrilateral from its properties?

Work through these questions in order:

1. 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).
2. Does it have exactly one pair of parallel sides? → trapezium.
3. Does it have two pairs of adjacent equal sides? → kite.
4. 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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