Quadrilaterals — Mathematics Knowledge System

Overview

A quadrilateral is a polygon with four vertices and four sides. Unlike a triangle, a quadrilateral is not rigid — its shape can flex even when all four side lengths are fixed, like a hinge — so classifying quadrilaterals requires extra conditions on angles, parallel sides, or diagonals. The family forms a nested hierarchy: every square is a rectangle and a rhombus, every rectangle and rhombus is a parallelogram, and every parallelogram is a trapezoid (under the inclusive definition). A foundational fact shared by all simple quadrilaterals is that their four interior angles always sum to 360°.

Intuition

Split any quadrilateral into two triangles by drawing one diagonal from a single vertex. Each triangle contributes 180° of interior angle, and together they account for exactly the quadrilateral's four angles, so the total is 2 × 180° = 360°. This diagonal-splitting trick is also why quadrilaterals are so useful for area: a parallelogram is literally a rectangle with a triangular sliver moved from one end to the other (same base, same height, same area), and any quadrilateral's area can be found by splitting it into two triangles and adding their areas.

Interactive Graph

Formal Definition

Definition

For a simple (non-self-intersecting) quadrilateral with interior angles α, β, γ, δ, and for a parallelogram with base b, height h, and sides a, b:

\alpha + \beta + \gamma + \delta = 360^\circ

Angle-sum theorem for quadrilaterals

A_{\text{parallelogram}} = bh

Area of a parallelogram (base times height)

A_{\text{trapezoid}} = \tfrac{1}{2}(b_1+b_2)h

Area of a trapezoid (average of parallel sides, times height)

A_{\text{rhombus}} = \tfrac{1}{2}d_1 d_2

Area of a rhombus from its diagonals

Notation

Notation	Meaning
$ABCD$	A quadrilateral named by its four vertices in order around the boundary
$d_1, d_2$	The two diagonals of a quadrilateral
$b_1, b_2$	The two parallel sides (bases) of a trapezoid

Properties

Quadrilateral angle-sum theorem

\alpha+\beta+\gamma+\delta = 360^\circ

Condition: Holds for every simple (non-self-intersecting) quadrilateral.

Example: A quadrilateral with angles 90°, 90°, 60°, 120° sums to exactly 360°.

Parallelogram properties

\text{Opposite sides parallel and equal; opposite angles equal; diagonals bisect each other.}

Condition: Defining property: both pairs of opposite sides are parallel.

Rectangle

\text{A parallelogram with four right angles; diagonals are equal in length.}

Rhombus

\text{A parallelogram with four equal sides; diagonals are perpendicular and bisect the vertex angles.}

Square

\text{A parallelogram that is both a rectangle and a rhombus: four equal sides and four right angles.}

Trapezoid (inclusive definition)

\text{A quadrilateral with at least one pair of parallel sides.}

Applications

Parallelogram linkages are used in mechanisms (e.g. pantographs, some car suspensions) because opposite sides stay parallel throughout the motion.

Worked Examples

The four interior angles sum to 360°.
$\delta = 360^\circ - 100^\circ - 80^\circ - 95^\circ = 85^\circ$

Answer: 85°

Practice Problems

Difficulty 2/10

A parallelogram has base 9 and height 4. Find its area.

Difficulty 3/10

A quadrilateral has angles 70°, 110°, and 95°. Find the fourth angle.

Difficulty 4/10

A trapezoidal plot of land has parallel edges 40 m and 60 m, with a perpendicular width (height) of 25 m between them. Find the area of the plot.

Common Mistakes

Common Mistake

Assuming every quadrilateral has an angle sum that depends on its shape, the way a triangle's does not vary.

Every simple quadrilateral, regardless of shape, has interior angles summing to exactly 360° — this follows from splitting it into two triangles via a diagonal.

Common Mistake

Thinking 'rhombus' and 'square' are unrelated categories.

A square is a special rhombus (one with right angles) and simultaneously a special rectangle (one with equal sides) — the quadrilateral families are nested, not disjoint.

Quiz

Summary

A quadrilateral has 4 sides and 4 vertices; its interior angles always sum to 360°.
Quadrilaterals form a nested hierarchy: square ⊂ rectangle, rhombus ⊂ parallelogram ⊂ trapezoid (inclusive definition).
Parallelogram area = bh; trapezoid area = ½(b₁+b₂)h; rhombus area = ½d₁d₂.
Unlike triangles, quadrilaterals are not rigid — fixed side lengths alone don't determine the shape.
Any quadrilateral's area can be computed by splitting it into two triangles via a diagonal.

References

WebsiteWikipedia — Quadrilateral