Quadrilaterals

Properties, Formulas, and Types of Quadrilaterals in Geometry

Overview of Quadrilaterals

A quadrilateral \(ABCD\) is a closed shape with four sides (edges) and four vertices. It is also referred to as a four-sided polygon.

quadrilateral

The four interior angles of a quadrilateral add up to \(360^\circ\), hence: \[ \angle A + \angle B + \angle C + \angle D = 360^\circ \]

Special quadrilaterals such as parallelograms, trapezoids, rectangles, squares, kites, and rhombuses are presented below with their specific properties and formulas.

Parallelogram

A parallelogram is a quadrilateral with both pairs of opposite sides parallel.

parallelogram

Properties

  1. Opposite sides are congruent (equal in length).
  2. The diagonals \(AC\) and \(BD\) bisect each other at point \(O\), dividing each diagonal into equal parts: \(OA = OC\) and \(OB = OD\).
  3. There are several pairs of congruent triangles: \(\triangle ADB \cong \triangle CBD\), \(\triangle ADC \cong \triangle ABC\), \(\triangle ADO \cong \triangle BCO\), etc.
  4. Opposite interior angles are congruent: \(\angle DAB = \angle DCB\) and \(\angle ADC = \angle ABC\).
  5. Several alternate/corresponding angles are congruent: \(\angle ABO = \angle CDO\), \(\angle DAO = \angle BCO\), etc.
  6. Area \(A_p\) of a parallelogram is given by: \[ A_p = b \times h \] where \(b\) is the base and \(h\) is the altitude.

Parallelogram Problems | Parallelogram Calculator

Trapezoid

A trapezoid is a quadrilateral with at least one pair of parallel opposite sides.

trapezoid

Area \(A_t\) of a trapezoid is given by: \[ A_t = \dfrac{1}{2} h \times (AB + DC) \]

Trapezoid Problems | Trapezoid Calculator

Rectangle

rectangle

Properties

A rectangle is a parallelogram with all interior angles equal to \(90^\circ\). Hence, all properties of parallelograms apply, along with additional properties listed below:

  1. Diagonals are equal in length and bisect each other: \(OA = OB = OC = OD\).
  2. Triangles formed by diagonals (\(\triangle ABC\), \(\triangle BCD\), etc.) are right triangles.
  3. Area \(A_{re}\) of a rectangle is given by: \[ A_{re} = AB \times BC \]

Rectangle Problems | Rectangle Calculator

Square

A square is a rectangle with all four sides equal in length.

square

Properties

All properties of rectangles and rhombuses apply to squares, with additional key features:

  1. The diagonals intersect at an angle of \(90^\circ\) (perpendicular).
  2. Area \(A_s\) of a square is given by: \[ A_s = AB^2 \]

Problems on Squares | Square Calculator

Kite

A kite is a quadrilateral with two pairs of adjacent sides equal in length. In the diagram below, sides \(BA\) and \(BC\) have equal lengths, and sides \(DA\) and \(DC\) have equal lengths.

kite

Properties

  1. The diagonals \(AC\) and \(BD\) intersect at point \(O\) at an angle of \(90^\circ\).
  2. Diagonal \(BD\) bisects diagonal \(AC\) at point \(O\), dividing \(AC\) into equal parts: \(OA = OC\).
  3. Triangles \(\triangle BOA\), \(\triangle BOC\), \(\triangle DOA\), and \(\triangle DOC\) are right triangles.
  4. \(\angle ABO = \angle CBO\), \(\angle ADO = \angle CDO\), etc.
  5. Triangles \(\triangle BOA\) and \(\triangle BOC\) are congruent, as are \(\triangle DOA\) and \(\triangle DOC\).
  6. Area \(A_k\) of a kite is given by: \[ A_k = \dfrac{1}{2} \times AC \times BD \]

Kite Questions with Solutions | Kite Calculator

Rhombus

A rhombus is a parallelogram with all 4 sides equal in length.

rhombus

Properties

  1. The diagonals \(AC\) and \(BD\) intersect at point \(O\) at an angle of \(90^\circ\) (perpendicular diagonals).
  2. The diagonals bisect each other at point \(O\), dividing both diagonals into equal parts: \(OA = OC\) and \(OD = OB\).
  3. Triangles \(\triangle BOA\), \(\triangle BOC\), \(\triangle DOA\), and \(\triangle DOC\) are congruent right triangles.
  4. \(\angle ABO = \angle CBO\), \(\angle ADO = \angle CDO\), etc.
  5. Area \(A_{ro}\) of a rhombus is given by: \[ A_{ro} = \dfrac{1}{2} \times AC \times BD \]

Rhombus Problems | Rhombus Calculator

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