Types of Quadrilaterals

A quadrilateral is any polygon with exactly four sides, four vertices, and four interior angles. The word comes from the Latin quadri (four) and latus (side). Quadrilaterals are everywhere in the built environment — windows, screens, floor tiles, and road signs — making them one of the most practically important figures in all of geometry. The sum of the interior angles of every quadrilateral, no matter how irregular, is always exactly 360°. Quadrilaterals are classified into a hierarchy based on which properties they possess. At the broadest level sits the general quadrilateral. Finer categories — trapezoids, parallelograms, rectangles, rhombuses, squares, and kites — are defined by increasingly specific conditions such as parallel sides, equal sides, or right angles. Understanding this hierarchy is essential for deductive reasoning: any property proven for a parallelogram is automatically inherited by every rectangle, rhombus, and square. In Euclidean geometry, proving that a figure belongs to a particular class requires a formal deductive argument using postulates, definitions, and previously established theorems. For example, proving a quadrilateral is a rectangle requires demonstrating not just that its angles look right but that they are rigorously 90° each. This article surveys the main types of quadrilaterals, their defining properties, and the key angle and side relationships that govern each one.

Key takeaways
  • General Quadrilateral
  • Trapezoid (Trapezium)
  • Parallelogram
  • Rectangle
  • Rhombus
  • Square
  • Kite

General Quadrilateral

A general (or irregular) quadrilateral has four sides and four angles with no special properties required. The only universal fact is that its interior angles sum to 360°. This follows by drawing a diagonal to split the quadrilateral into two triangles, each contributing 180°, giving 2 × 180° = 360° total.

Trapezoid (Trapezium)

A trapezoid has exactly one pair of parallel sides, called the bases. The non-parallel sides are called legs. If the legs are equal in length, the figure is an isosceles trapezoid, which has equal base angles and a line of symmetry. The midsegment (median) of a trapezoid connects the midpoints of the legs and has length equal to the average of the two bases: m = (b₁ + b₂)/2. Note: some curricula define a trapezoid as having at least one pair of parallel sides, which makes parallelograms a special case.

Parallelogram

A parallelogram has two pairs of parallel sides. This single condition produces a cascade of consequences: opposite sides are equal in length, opposite angles are equal, and consecutive angles are supplementary (sum to 180°). The diagonals of a parallelogram bisect each other — they cut each other into two equal segments at their intersection point. Any quadrilateral satisfying any one of these equivalent conditions (opposite sides parallel, opposite sides equal, diagonals bisecting each other) can be proven to be a parallelogram.

Rectangle

A rectangle is a parallelogram in which all four interior angles are right angles (90°). Since it is a parallelogram, opposite sides remain equal and parallel. Rectangles have the additional property that their diagonals are equal in length. If a rectangle has side lengths a and b, its diagonal d satisfies d = √(a² + b²) by the Pythagorean theorem, and its area is A = ab.

Rhombus

A rhombus is a parallelogram in which all four sides are equal in length. It is sometimes called an equilateral quadrilateral. The diagonals of a rhombus are perpendicular bisectors of each other — they meet at right angles and each bisects the other. They also bisect the vertex angles. If the diagonals have lengths d₁ and d₂, the area of a rhombus is A = (d₁ × d₂)/2.

Square

A square is simultaneously a rectangle and a rhombus: all four sides are equal and all four angles are 90°. It is the most symmetric quadrilateral, possessing four lines of symmetry and rotational symmetry of order 4. For a square with side length s, the diagonal length is d = s√2, the perimeter is 4s, and the area is s².

Kite

A kite has two pairs of consecutive (adjacent) sides that are equal. Unlike a parallelogram, the equal sides are next to each other, not opposite. One diagonal of a kite is the perpendicular bisector of the other. The area of a kite with perpendicular diagonals d₁ and d₂ is A = (d₁ × d₂)/2, the same formula as the rhombus. A square is a special case of both a rhombus and a kite.

The Key Equation

∑i=14∠i=360°\sum_{i=1}^{4} \angle_i = 360°
  • \angle_i — Interior angle i (degrees): Each of the four interior angles of the quadrilateral, numbered 1 through 4.
  • 360° — Angle sum (degrees): The constant total interior angle measure for all quadrilaterals, derived by splitting the figure into two triangles each having angle sum 180°.

Worked Example

Problem. In parallelogram ABCD, angle A = (3x + 10)° and angle B = (5x − 30)°. Find the value of x and all four angles of the parallelogram.

x=25,∠A=∠C=85°,∠B=∠D=95°x = 25,\quad \angle A = \angle C = 85°,\quad \angle B = \angle D = 95°

Answer. x = 25; Angle A = Angle C = 85°; Angle B = Angle D = 95°

Figure
Diagram
Quadrilateral Classification HierarchyQuadrilateralTrapezoidParallelogramKiteRectangleRhombusSquareTrapezoidParallelogramRectangleSquare
Origins · 1 minEuclid of Alexandria and how this was discovered
EO
Euclid of Alexandria
-300–-260 · Greek

Euclid systematized plane geometry in his 13-volume work Elements (c. 300 BCE). Book I contains the foundational theorems on triangles and quadrilaterals, including the proof that the interior angles of a triangle sum to 180° — the result that directly implies the 360° angle sum for quadrilaterals. He defined parallelograms (calling them parallelogrammic areas) and proved their key properties about opposite sides, opposite angles, and diagonal bisection, establishing the deductive framework that still underlies high-school geometry today. Euclid's Elements was used as the standard geometry textbook in Western education for over 2,000 years — from ancient Greece through the late 19th century — making it one of the most influential textbooks ever written.

  • Triangle Angle Sum Theorem — The proof that quadrilateral angles sum to 360° depends directly on the triangle angle sum of 180°, since any quadrilateral can be divided into two triangles by a diagonal.
  • Properties of Parallelograms — Parallelograms form the central branch of the quadrilateral hierarchy; rectangles, rhombuses, and squares are all special parallelograms with additional properties.
  • Pythagorean Theorem — The diagonal length of a rectangle (d = √(a² + b²)) and square (d = s√2) are computed directly using the Pythagorean theorem.
  • Area of Quadrilaterals — Each type of quadrilateral has its own area formula derived from its specific properties, such as A = bh for parallelograms and A = (d₁d₂)/2 for rhombuses and kites.
  • Deductive Reasoning in Geometry — Proving a quadrilateral belongs to a specific class (e.g., showing a shape is a parallelogram) is a key application of formal two-column and paragraph proofs.