Sorting Shapes into Categories

Shapes are sorted by what they have

A shape goes into a group because of its sides and corners. Nothing else counts.

Color does not count. Size does not count. Which way up it sits does not count.

A small red triangle and a huge blue triangle are both triangles. Each has three straight sides, and that is the whole test.

Shape	Straight sides
triangle	$3$
quadrilateral	$4$
pentagon	$5$
hexagon	$6$

Quadrilateral means four sides

A quadrilateral is any closed shape with four straight sides.

That is a big group. It holds shapes that look nothing like each other.

All three have four sides. So all three are quadrilaterals.

The group is decided by the count of sides, not by the look of the shape.

Why one shape sits in several groups

Inside quadrilaterals there are smaller groups, and a shape can be in more than one of them.

Group	What it needs
rectangle	four square corners
rhombus	four equal sides
square	four square corners and four equal sides

Look at the square row. It has everything a rectangle needs. It also has everything a rhombus needs.

So a square is a rectangle, a rhombus and a quadrilateral, all at once.

$$\text{square} \;\subset\; \text{rectangle} \;\subset\; \text{quadrilateral}$$

This is what makes sorting useful. A rule proved for every rectangle is proved for every square too, because squares are already inside that group. You never have to prove it twice.

Going the other way does not work. A long rectangle has square corners but not equal sides, so it is not a square.

Drawing one that fits no small group

Some quadrilaterals have four sides and nothing else special. No square corners. No equal sides.

The third shape in the picture above is one of these. It is a quadrilateral and nothing more.

Drawing one is a good check. Make four straight sides, join them up, and keep every side a different length.

Splitting a shape into equal parts

A shape can be cut into parts that each cover the same amount.

Cut a rectangle into $4$ equal parts. Each part is one quarter of it:

$$\frac{1}{4}$$

The bottom number is the count of parts. This is the same unit fraction idea, used on a shape instead of a pizza.

Equal parts	Each part is
$2$	$\tfrac{1}{2}$
$3$	$\tfrac{1}{3}$
$4$	$\tfrac{1}{4}$
$6$	$\tfrac{1}{6}$

The parts do not have to be the same shape. They have to cover the same amount. A square cut corner to corner gives two triangles, and each is still $\tfrac{1}{2}$ of the square.

Worked examples

Common mistakes

Practice problems