Four-Sided Shapes

A door, a page of this book, a floor tile, a kite flying on a windy beach, a slice of cake cut from a long tray. All of them have four sides. But they don't look the same. How do you tell these four-sided shapes apart, and how are they related?

The family of quadrilaterals

A polygon with four sides is a quadrilateral. Some quadrilaterals are special enough to have their own names. Each name says what is special about the sides or the angles:

a rectangle has four right angles;
a square has four right angles and four equal sides;
a parallelogram has two pairs of parallel sides: the top and bottom are parallel, and so are the left and right;
a rhombus has four equal sides (it looks like a pushed-over square);
a trapezium has just one pair of parallel sides (in the United States it is called a trapezoid);
a kite has two pairs of equal sides that sit next to each other, like the toy.

Six quadrilaterals with their names: a rectangle and a square with right angles marked; a parallelogram with arrow marks on its two pairs of parallel sides; a rhombus with marks showing its four equal sides; a trapezium with one pair of parallel sides; and a kite with marks showing two pairs of equal sides next to each other

A square is a rectangle too

Is a square a rectangle? Read what a rectangle needs: four right angles. A square has four right angles, so yes, a square is a rectangle: a special one whose sides are all equal. In the same way, a square is a rhombus (four equal sides), and every rectangle and every rhombus is a parallelogram (both pairs of opposite sides are parallel).

So the shapes form a family tree. The higher a name is, the more shapes it covers; the lower, the more it demands.

A family tree of shapes. At the top, quadrilateral. Below it, trapezium, parallelogram and kite. Below parallelogram, rectangle and rhombus. At the bottom, square, joined to both rectangle and rhombus

It works like the names of animals: every cat is an animal, but not every animal is a cat. Every square is a rectangle, but not every rectangle is a square.

Diagonals

A line segment from one vertex to the opposite vertex is called a diagonal. Every quadrilateral has two of them. A diagonal cuts a quadrilateral into two triangles. In the last lesson we saw that the angles of each triangle add up to 180°. Together, the two triangles have exactly the angles of the quadrilateral, so the angles of every quadrilateral add up to 180° + 180° = 360°. A rectangle checks it: 4 × 90° = 360°.

A four-sided shape with one diagonal drawn from corner to corner, cutting it into two triangles, one shaded blue and one shaded amber. Each triangle's angles add up to 180°, so the four angles together make 360°

Try it at home

Find a rectangle, a square and, if you can, a trapezium in your home.
Is every rectangle a square?
Three angles of a quadrilateral are 90°, 90° and 100°. How big is the fourth?

Answers: 2. No: a rectangle's sides don't have to be equal. A square is a special rectangle. 3. 90° + 90° + 100° = 280°, and 360° - 280° = 80°.

Words to know

quadrilateral - a polygon with four sides
rectangle - four right angles; square - four right angles and four equal sides
parallelogram - two pairs of parallel sides; rhombus - four equal sides
trapezium - one pair of parallel sides; kite - two pairs of equal sides next to each other
diagonal - a line segment joining two opposite vertices

Big idea

Quadrilaterals form a family, each name saying what is special about the sides or angles. A square is a rectangle and a rhombus at the same time. A diagonal cuts any quadrilateral into two triangles, so its angles add up to 360°.