Quadrilaterals: Types, Properties, and Area Formulas

A quadrilateral is a polygon with four sides, four vertices, and four interior angles that always add up to $360^\circ$ . The main types are the parallelogram, rectangle, square, rhombus, trapezoid, and kite, and each has its own property set and area formula.

What is a quadrilateral?

Any closed figure made of four straight line segments is a quadrilateral. Drawing one diagonal splits it into two triangles, and since each triangle has an angle sum of $180^\circ$ , the angle sum of every quadrilateral is

180^\circ + 180^\circ = 360^\circ

This single fact solves a huge share of quadrilateral angle problems: if three angles are known, the fourth is always $360^\circ$ minus their sum.

Types of quadrilaterals and their properties

The named quadrilaterals form a hierarchy. A square is a special rectangle and a special rhombus; rectangles and rhombuses are special parallelograms.

Type	Sides	Angles	Diagonals
Parallelogram	Opposite sides parallel and equal	Opposite angles equal	Bisect each other
Rectangle	Opposite sides parallel and equal	All angles $90^\circ$	Equal, bisect each other
Square	All four sides equal	All angles $90^\circ$	Equal, bisect at $90^\circ$
Rhombus	All four sides equal	Opposite angles equal	Bisect at $90^\circ$
Trapezoid	Exactly one pair of parallel sides	Co-interior angles on each leg sum to $180^\circ$	No special rule in general
Kite	Two pairs of adjacent equal sides	One pair of opposite angles equal	Perpendicular; one bisects the other

Two quick consequences worth memorizing:

In a parallelogram, consecutive angles are supplementary: $\angle A + \angle B = 180^\circ$ .
In a kite or rhombus, the diagonals cross at right angles, which is why their areas use diagonals.

Area formulas for every quadrilateral

Quadrilateral	Area formula	Variables
Square	$A = a^2$	$a$ = side
Rectangle	$A = l \times w$	length, width
Parallelogram	$A = b \times h$	base, perpendicular height
Rhombus	$A = \tfrac{1}{2} d_1 d_2$	the two diagonals
Trapezoid	$A = \tfrac{1}{2}(b_1 + b_2)h$	parallel sides, height
Kite	$A = \tfrac{1}{2} d_1 d_2$	the two diagonals

In every formula, $h$ means the perpendicular height — never a slanted side. The rhombus and kite share the same diagonal formula because both have perpendicular diagonals.

Worked example 1: area of a rhombus from its diagonals

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

Area first:

A = \frac{1}{2} d_1 d_2 = \frac{1}{2}(10)(24) = 120\ \text{cm}^2

For the side, the diagonals bisect each other at right angles, so each side is the hypotenuse of a right triangle with legs $5$ cm and $12$ cm:

s = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13\ \text{cm}

So the rhombus has area $120\ \text{cm}^2$ and side $13$ cm.

Worked example 2: missing angle in a quadrilateral

A quadrilateral has angles $x$ , $2x$ , $95^\circ$ , and $85^\circ$ . Find $x$ .

Use the angle sum:

x + 2x + 95^\circ + 85^\circ = 360^\circ

Combine like terms:

3x + 180^\circ = 360^\circ \quad\Rightarrow\quad 3x = 180^\circ \quad\Rightarrow\quad x = 60^\circ

The four angles are $60^\circ$ , $120^\circ$ , $95^\circ$ , and $85^\circ$ . A quick check: $60 + 120 + 95 + 85 = 360$ . ✓

Common mistakes with quadrilaterals

Using a slanted side as the height in parallelogram or trapezoid area.
Treating every four-sided shape as a parallelogram — a general quadrilateral has no equal-opposite-sides rule.
Multiplying the full diagonals without the $\tfrac{1}{2}$ in rhombus and kite areas.
Assuming a rhombus has right angles; only the square does.
Forgetting that "square is a rectangle" is true but "rectangle is a square" is not — hierarchy questions test this exact direction.

Where quadrilateral properties show up next

Quadrilaterals carry directly into coordinate geometry, where you prove a figure is a parallelogram by comparing slopes or midpoints of diagonals. They also anchor circle theorems: a cyclic quadrilateral has opposite angles summing to $180^\circ$ . If you are comfortable with the property table above, both of those later topics become checks of facts you already know rather than new memorization.

Frequently Asked Questions

What are the 6 types of quadrilaterals?: The six standard types are the parallelogram, rectangle, square, rhombus, trapezoid, and kite. They form a hierarchy: a square is both a special rectangle and a special rhombus, while rectangles and rhombuses are special parallelograms. A trapezoid has exactly one pair of parallel sides, and a kite has two pairs of adjacent equal sides.
What is the sum of angles in a quadrilateral?: The interior angles of any quadrilateral always add up to 360 degrees. This follows from splitting the quadrilateral into two triangles with one diagonal, each contributing 180 degrees. If three angles are known, subtract their total from 360 to find the fourth angle.
What is the area formula for each quadrilateral?: Square: side squared. Rectangle: length times width. Parallelogram: base times perpendicular height. Trapezoid: half the sum of the parallel sides times the height. Rhombus and kite: half the product of the two diagonals. In every formula the height must be perpendicular, not a slanted side.
Is a square a rectangle?: Yes. A square satisfies every rectangle property: opposite sides parallel and equal, and all four angles 90 degrees. It simply adds the extra condition that all sides are equal. The reverse is not true, since a rectangle does not need equal adjacent sides.
What is the difference between a rhombus and a parallelogram?: Every rhombus is a parallelogram, but a rhombus additionally has all four sides equal and diagonals that cross at right angles. A general parallelogram only requires opposite sides to be equal and parallel, and its diagonals bisect each other without being perpendicular.

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