Quadrilaterals – Four Sides, Many Shapes

A quadrilateral is any closed shape with four straight sides and four angles. Quadrilaterals are everywhere: the rooms we live in, the screens we look at, and the tiles on the floor are all quadrilaterals.

The word “quadrilateral” comes from the Latin quadri (four) and latus (side), and Euclid devoted much of Book I of his Elements (c. 300 BCE) to proving properties of parallelograms — for instance, that a diagonal splits a parallelogram into two triangles of equal area, a result he needed on the way to proving Pythagoras’ theorem itself. One quirky bit of quadrilateral trivia: the shape with exactly one pair of parallel sides is called a trapezium in British English, but in American English “trapezium” instead means an irregular four-sided shape with no parallel sides at all (called a trapezoid in the UK) — the two dialects swapped the definitions, which still causes confusion in international textbooks today. Beyond the classroom, quadrilaterals dominate the built world: door frames, book covers, solar panels, and city blocks are almost always rectangles or parallelograms, chosen because their straight, parallel edges tile and stack together without leaving gaps.

Key Properties of All Quadrilaterals

Every quadrilateral has four sides, four vertices, and two diagonals. The interior angles of any quadrilateral always sum to 360°.

Types of Quadrilaterals

Shape	Sides	Angles	Diagonals
Square	All 4 equal; all parallel in pairs	All 90°	Equal, bisect at 90°
Rectangle	Opposite sides equal; all parallel in pairs	All 90°	Equal, bisect each other
Rhombus	All 4 equal; opposite sides parallel	Opposite angles equal	Bisect at 90°, not equal
Parallelogram	Opposite sides equal and parallel	Opposite angles equal	Bisect each other
Trapezium	One pair of parallel sides	Co-interior angles sum to 180°	Not equal, do not bisect
Kite	Two pairs of adjacent equal sides	One pair of equal opposite angles	One bisects the other at 90°

Angle Sum Property

A quadrilateral has three angles of 85°, 100°, and 95°. Find the fourth.

Fourth angle = 360 − 85 − 100 − 95 = 80°.

A parallelogram has one angle of 65°. Find all four angles.

Opposite angles equal: two angles are 65°. Adjacent angles are supplementary: 180 − 65 = 115°. Angles: 65°, 115°, 65°, 115°.

Areas of Quadrilaterals

Shape	Area Formula
Square	side²
Rectangle	length × width
Parallelogram	base × perpendicular height
Rhombus	½ × d₁ × d₂ (product of diagonals)
Trapezium	½ × (a + b) × h
Kite	½ × d₁ × d₂

Key Takeaways

All quadrilaterals have four sides and interior angles summing to 360°.
A square is the most special: all sides equal AND all angles 90°.
A rectangle has all 90° angles but sides need not all be equal.
A rhombus has all equal sides but angles need not be 90°.
A trapezium has exactly one pair of parallel sides.

Practice: Angle Sum & Area

Quadrilateral Angle Sum