Is a Square a Quadrilateral? Polygons Explained

Polygons and quadrilaterals: is a square a quadrilateral?

Is a square a quadrilateral? Yes — and a rectangle, a rhombus and a parallelogram. How the family fits together, plus polygon angle sums and areas.

1 min readUpdated 2 November 2026By Sumzo team (AI-assisted)Reviewed by Sumzo AI (owner-delegated review)

Is a square a quadrilateral? Yes. A quadrilateral is any polygon with four straight sides, and a square has four straight sides — so every square is a quadrilateral. The reverse isn’t true: a quadrilateral is not always a square. A square is the most special member of the quadrilateral family.

The quadrilateral family

Quadrilateral: any four-sided polygon.
Trapezoid (trapezium in the UK): at least one pair of parallel sides (some textbooks say exactly one).
Parallelogram: two pairs of parallel sides.
Rectangle: a parallelogram with four right angles.
Rhombus: a parallelogram with four equal sides.
Square: both a rectangle and a rhombus — four right angles and four equal sides.

So a square is a quadrilateral, a parallelogram, a rectangle and a rhombus all at once. A rectangle is a square only if its sides are equal.

Angles in polygons

\text{interior angle sum} = (n - 2) \times 180^\circ

For a polygon with $n$ sides.

Worked example

Easy: the angle sum of a hexagon

Find the sum of the interior angles of a hexagon.

Show the working

A hexagon has 6 sides. $(6 - 2) \times 180^\circ = 720^\circ$

Answer: $720^\circ$

Checked by Sumzo’s math engine

Worked example

Medium: one angle of a regular octagon

Find each interior angle of a regular octagon.

Show the working

Angle sum: $(8 - 2) \times 180° = 1080°$ .
All 8 angles are equal. $8x = 1080 \Rightarrow x = 135^\circ$

Answer: $135^\circ$

Checked by Sumzo’s math engine

Areas of quadrilaterals

Worked example

Medium: a parallelogram

Find the area of a parallelogram with base 9 cm and perpendicular height 4 cm.

Show the working

Area = base × perpendicular height. $9 \times 4 = 36\text{ cm}^2$

Answer: $36\text{ cm}^2$

Checked by Sumzo’s math engine

Worked example

Hard: a trapezoid

Find the area of a trapezoid with parallel sides 5 cm and 9 cm and height 4 cm.

Show the working

Half the sum of the parallel sides, times the height. $\tfrac{1}{2}(5 + 9) \times 4 = 28\text{ cm}^2$

Answer: $28\text{ cm}^2$

Checked by Sumzo’s math engine

Common mistakes

Thinking a square is not a rectangle — it is a rectangle with equal sides.
Using the slanted side of a parallelogram as its height.
Using $n \times 180°$ instead of $(n - 2) \times 180°$ for the angle sum.
Calling every four-sided shape with a right angle a rectangle.

Frequently asked questions