Types of Quadrilateral

Four straight sides

Look around a room and nearly every flat shape you see — the floor tiles, the tabletop, a road sign, a picture frame — has four straight sides. Builders and designers name these four-sided shapes carefully, because calling something a square rather than a rhombus pins down exactly which sides and angles you can count on.

A quadrilateral is any flat shape made of four straight sides joined up into a closed loop. "Quad" means four and "lateral" means side — so the word literally says four sides. A triangle has three, a quadrilateral has four, and that one extra side gives us a whole family of new shapes.

Every quadrilateral, no matter how squashed or stretched, has four corners. And here is a fact that is always true: those four corner angles add up to exactly 360^\circ. That is the same as turning all the way around a full circle. Cut any quadrilateral into two triangles along a diagonal — each triangle's angles make 180^\circ, and 180^\circ + 180^\circ = 360^\circ.

bus

Once you start looking you cannot stop seeing them. A bus is full of quadrilaterals: the windows and the doors are rectangles, and so is the screen you are reading this on. A book, a door, a football pitch, a chessboard, a slice of toast — all four-sided. Hunt for ten quadrilaterals in the room around you right now.

The six quadrilaterals

There are six four-sided shapes you meet again and again, each pinned down by one defining property:

a square has four equal sides and four right angles;
a rectangle has four right angles, with opposite sides equal;
a rhombus has four equal sides (a "pushed-over" square);
a parallelogram has both pairs of opposite sides parallel;
a trapezium has exactly one pair of parallel sides;
a kite has two pairs of adjacent equal sides.

Mathematicians draw these properties straight onto the shape so you can read them at a glance. Equal sides get matching tick marks (one tick matches one tick, two ticks match two ticks); parallel sides get matching arrowheads; and a right angle gets a small square in the corner.

Meet the family

Step through the gallery to see one of each, drawn with its defining marks. Watch how the marks change from shape to shape — they are exactly the property in each name.

How they fit together

These shapes are not six separate boxes — the special ones are particular cases of the more general ones. Adding a rule (equal sides, or right angles) to a parallelogram is what creates the squares, rectangles and rhombuses.

every quadrilateral's interior angles sum to 360^\circ;
a square is a special rectangle (it just also has equal sides) and a special rhombus (it just also has right angles);
a rectangle and a rhombus are both special parallelograms — once opposite sides are equal, they are automatically parallel.

The family only works one way:

every square is a rectangle (it has four right angles), but not every rectangle is a square — a long thin rectangle has no equal sides.
a rhombus is just a "pushed-over" square: tilt a square so its corners stop being right angles and you get a rhombus. It keeps its four equal sides but loses its right angles.
a square is the only shape that is a rectangle and a rhombus and a parallelogram all at once — it has every property.

balloon on a string

The geometry kite is named after the toy that flies on a string. A real kite has two short sides at the top meeting two long sides at the bottom — two pairs of neighbouring equal sides. That is different from a rhombus, where all four sides match. Next time you see something flying on a string, check its corners: is it a true kite shape?

Name that shape

Here is a quadrilateral drawn with its marks but you have to read them. Look at the ticks, the arrowheads and any corner squares, then check the name underneath. Press Refresh for a fresh shape and try to call out its name before you read it.

Worked examples

1. Is a square a rectangle?

A rectangle's rule is "four right angles". A square has four right angles — so it passes the test. Yes, every square is a rectangle; it simply has the bonus that all four sides are equal too. (But a rectangle is only a square when its sides happen to be equal.)

2. Three corners of a quadrilateral are 100^\circ, 80^\circ and 90^\circ. What is the fourth?

The four angles must total 360^\circ, so the missing angle is 360^\circ - 100^\circ - 80^\circ - 90^\circ = 90^\circ.

3. A shape has four equal sides but no right angles. What is it?