Math Core

Scientists sort animals into groups, like mammals and birds, based on what they have in common. You can sort flat shapes the same way, using their sides and angles. Once you know the groups, you can give any shape every name that fits it.

Polygons

A polygon is a closed flat shape made of straight sides. Polygons are named by how many sides they have: a triangle has 3, a quadrilateral has 4, a pentagon has 5, a hexagon has 6 and an octagon has 8. A circle is not a polygon, because it has no straight sides.

Sorting triangles

You can sort triangles by their sides:

Equilateral: all 3 sides are the same length.
Isosceles: at least 2 sides are the same length.
Scalene: no sides are the same length.

Left to right: an equilateral, an isosceles and a scalene triangle.

You can also sort triangles by their angles:

Right: one angle is a right angle ( $90^\circ$ ).
Acute: all 3 angles are less than $90^\circ$ .
Obtuse: one angle is greater than $90^\circ$ .

Left to right: a right, an acute and an obtuse triangle.

Every triangle gets one name from each list. A triangle with a right angle and two equal sides is a right isosceles triangle.

The quadrilateral family

Quadrilaterals are sorted by their parallel sides, equal sides and right angles.

Shape	What it needs
trapezoid	at least 1 pair of parallel sides
parallelogram	2 pairs of parallel sides
rectangle	a parallelogram with 4 right angles
rhombus	a parallelogram with 4 equal sides
square	a parallelogram with 4 right angles and 4 equal sides

Left to right: a parallelogram, a trapezoid and a rhombus.

Some books say a trapezoid has exactly one pair of parallel sides. This lesson uses "at least one pair."

A hierarchy of shapes

The groups fit inside each other, like boxes inside boxes. This is called a hierarchy.

Every square is a rectangle and a rhombus.
Every rectangle and every rhombus is a parallelogram.
Every parallelogram is a quadrilateral.
Every quadrilateral is a polygon.

Attributes are passed down

If a shape belongs to a group, it has every attribute of that group and of all the bigger groups above it. A square has 2 pairs of parallel sides because it is a parallelogram, and 4 sides because it is a quadrilateral.

The hierarchy only works in one direction. Every square is a rectangle, but not every rectangle is a square.

Worked example: All the names

A shape has 4 sides, 2 pairs of parallel sides and 4 right angles. Its sides are 6 cm, 3 cm, 6 cm and 3 cm. List every name that fits.

It has 4 sides, so it is a quadrilateral (and a polygon). It has 2 pairs of parallel sides, so it is a parallelogram. It has 4 right angles, so it is a rectangle. Its sides are not all equal, so it is not a rhombus or a square.

Worked example: True or false?

"Every rhombus is a rectangle." Is this true?

False. A rhombus needs 4 equal sides, but it does not need right angles. The rhombus in the picture above has no right angles, so it is not a rectangle.

Common mistake

A shape's name doesn't change when you turn it. A square standing on one corner still has 4 equal sides and 4 right angles, so it is still a square.

Practice

Practice 1

A polygon has 6 sides. What is it called?

Practice 2

A triangle has sides that are 5 cm, 8 cm and 5 cm long. How is it classified by its sides?

Practice 3

One angle of a triangle measures $110^\circ$ . How is the triangle classified by its angles?

Practice 4

Which property do all parallelograms have?

Practice 5

Which statement is true?

Practice 6

How many of these names describe a square?

polygon, triangle, quadrilateral, parallelogram, rectangle, rhombus

Type a number.

Practice 7

An equilateral triangle has a perimeter of 27 inches. How long is each side, in inches?

Type a number.

Practice 8

Jada says, "My shape is a rhombus, but it is not a square." Which must be true about her shape?