Quadrilaterals — Types, Properties, and Formulas

Quadrilaterals — Types, Properties, and Formulas

A quadrilateral is a closed, four-sided polygon whose four interior angles always add up to 360°. This guide maps the whole family (parallelogram, rectangle, square, rhombus, trapezium, and kite), showing how each one inherits or breaks the rules of the others, with their diagonals, area formulas, and worked examples.

What Is A Quadrilateral?

A quadrilateral is a flat shape with four straight sides, four vertices, and four angles. The four interior angles always sum to 360°. The name comes from the Latin quadri (four) and latus (side), and any closed figure you can draw with four line segments, from a perfect square to a lopsided arrowhead, belongs to this family.

Two facts hold for every quadrilateral, no matter how stretched or skewed: it has exactly two diagonals (line segments joining opposite corners), and its angles total 360°. Everything else, such as equal sides, parallel sides, or right angles, is what splits the family into the named types below.

Why Does The Angle Sum Equal 360°?

Here is the cleanest way to see it. Draw any one diagonal of a quadrilateral, say from A to C, and that single line splits the four-sided shape into two triangles. The angles inside a triangle always add to 180°, a result that traces back to Euclid's Elements. Two triangles give 2×180°=360°.

This is the engine behind the whole family. When a shape forces some angles to be 90°, the others are pinned down too, which is why a rectangle's "fourth" angle is never a free choice.

What Are The Six Types of Quadrilaterals?

There are six named quadrilaterals every student meets. The trick is to stop memorising them as six separate shapes and see them as a hierarchy — each one is a more constrained version of a looser one.

A square satisfies every rule at once, which is why it sits at the bottom of the hierarchy: every square is a rectangle, every square is a rhombus, and every rectangle, rhombus, and square is a parallelogram. The reverse is never guaranteed: a rectangle is only a square when its sides happen to be equal.

How Are Quadrilaterals Classified (convex vs concave)?

Before the named types, quadrilaterals split into two broad groups by shape.

A convex quadrilateral has every interior angle less than 180°, and both diagonals sit inside the figure. All six named types above are convex. A concave quadrilateral (sometimes called a re-entrant or arrowhead quadrilateral) has one interior angle greater than 180°, and one diagonal falls outside the shape. There is also a third, rarer case, the complex or crossed quadrilateral, where two sides intersect, but it shows up far less often at this level.

A useful check: if you can connect every pair of corners with a diagonal that stays inside the shape, it is convex. If one diagonal escapes outside, it is concave.

Properties of Quadrilaterals at a Glance

Type Sides Angles Diagonals
Parallelogram Opposite sides equal and parallel Opposite angles equal Bisect each other
Rectangle Opposite sides equal All four are 90° Equal, bisect each other
Square All four equal All four are 90° Equal, bisect at 90°
Rhombus All four equal Opposite angles equal Bisect each other at 90°
Trapezium One pair parallel Co-interior angles on each leg sum to 180° Not generally equal
Kite Two adjacent pairs equal One pair of opposite angles equal One bisects the other at 90°

Area And Perimeter Formulas For Each Quadrilateral

The area formulas look unrelated until you notice they all come from the same idea: chop the shape into pieces you already know how to measure. Here is each formula with what every symbol means.

Square, side s: Area = s², Perimeter = 4s

Rectangle, length l and width w: Area = l×w, Perimeter = 2(l + w)

Parallelogram, base b and perpendicular height h: Area = b×h

Rhombus, diagonals d₁ and d₂: Area = 1/2 × d₁ × d₂

Trapezium, parallel sides a and b with height h: Area = 1/2 × (a + b) × h

Kite, diagonals d₁ and d₂: Area = 1/2 × d₁ × d₂

The rhombus and kite share a formula because both have diagonals that meet at a right angle, and any shape whose diagonals are perpendicular has area equal to half their product.

Examples of Quadrilaterals

Example 1

A quadrilateral has three angles measuring 80°, 95°, and 100°. Find the fourth angle.

The four angles sum to 360°. 80° + 95° + 100° = 275°
Fourth angle = 360° - 275° = 85°
Final answer: 85°.

Example 2

Identify the most specific name for a quadrilateral with all four sides equal and all four angles equal to 90°.

The shape that demands equal sides and four right angles is the square. Final answer: Square.

Example 3

Find the area of a parallelogram with base 12 cm and perpendicular height 7 cm.

Area = b × h = 12 × 7 = 84 cm²
Final answer: 84 cm².

Example 4

A rhombus has diagonals of 10 cm and 24 cm. Find its area.

Area = 1/2 × d₁ × d₂ = 1/2 × 10 × 24 = 120 cm²
Final answer: 120 cm².

Example 5

A trapezium has parallel sides of 8 m and 14 m, with a height of 5 m. Find its area.

Area = 1/2 × (8 + 14) × 5 = 55 m²
Final answer: 55 m².

Example 6

A kite has diagonals measuring 16 cm and 9 cm. Find its area, then explain why the same formula works for a rhombus.

Area = 1/2 × d₁ × d₂ = 1/2 × 16 × 9 = 72 cm²
Final answer: 72 cm².

Where The Quadrilateral Family Earns Its Keep

Quadrilaterals are not a syllabus invention. The classification exists because the rules a shape obeys decide what it can do.

Conclusion