Quadrilaterals

What is a Quadrilateral?

A quadrilateral is any polygon with four sides and four angles. The sum of interior angles in any quadrilateral is always 360°.

You can split any quadrilateral into two triangles by drawing one diagonal — since each triangle has angle sum 180°, the total is 2 × 180° = 360°.

Key Terms

• Parallel sides: sides that are always the same distance apart and never meet
• Bisect: to cut exactly in half
• Diagonal: a line segment connecting two non-adjacent vertices
• Line of symmetry: a line that divides a shape into two mirror-image halves
• Perpendicular: meeting at a right angle (90°)

Properties at a Glance

Square

• All 4 sides equal
• All angles = 90°
• 4 lines of symmetry
• Diagonals are equal, bisect each other at right angles, and bisect the corner angles

Area = side²

Rectangle

• Opposite sides equal and parallel
• All angles = 90°
• 2 lines of symmetry (horizontal and vertical)
• Diagonals are equal in length and bisect each other (but do NOT cross at right angles)

Area = length × width

Parallelogram

• Opposite sides equal and parallel
• Opposite angles equal (co-interior angles add to 180°)
• No lines of symmetry in general
• Diagonals bisect each other but are not equal and not perpendicular

Area = base × perpendicular height

Note: a rectangle is a special parallelogram where all angles are 90°.

Rhombus

• All 4 sides equal (like a "squished" square)
• Opposite angles equal
• 2 lines of symmetry (along the diagonals)
• Diagonals bisect each other at right angles, and bisect the corner angles
• Diagonals are NOT equal in length (unlike a square)

Area = (diagonal 1 × diagonal 2) / 2

Trapezium

• Exactly one pair of parallel sides (called the parallel sides or bases)
• The other two sides are called the legs
• An isosceles trapezium has equal legs and one line of symmetry

Area = ½(a + b) × h

where a and b are the parallel sides and h is the perpendicular height.

Worked example: Parallel sides of 6cm and 10cm, height 4cm: Area = ½(6 + 10) × 4 = ½ × 16 × 4 = 32 cm²

Kite

• Two pairs of adjacent (next to each other) sides equal
• One pair of opposite angles equal (the angles between the unequal sides)
• 1 line of symmetry (along the longer diagonal)
• Diagonals are perpendicular; the longer diagonal bisects the shorter one

Area = (diagonal 1 × diagonal 2) / 2

Hierarchy of Quadrilaterals

Quadrilaterals form a family with a hierarchy based on shared properties:

• Every square is also a rectangle, a rhombus, and a parallelogram
• Every rectangle is a parallelogram
• Every rhombus is a parallelogram
• A parallelogram is NOT necessarily a rectangle or rhombus

This means any property true of all parallelograms is also true of squares, rectangles, and rhombuses.

Common Mistakes to Avoid

• Confusing rhombus and square: a rhombus has 4 equal sides but angles are NOT necessarily 90°; a square has both
• Calling a trapezium a parallelogram: a trapezium has only ONE pair of parallel sides, not two
• Incorrect trapezium area: do not forget to halve — Area = ½(a + b) × h, not just (a + b) × h
• Thinking a kite has two pairs of opposite equal sides: it has two pairs of adjacent equal sides

Tips and Tricks

• Learn the hierarchy: Square → Rectangle → Parallelogram → Quadrilateral helps remember shared properties
• Use the angle sum (360°) to find missing angles when three are known
• For area of a rhombus or kite, the diagonal formula is often easier than base × height
• A parallelogram and a rectangle with the same base and height have the same area