In Grade 4 you sorted shapes by their sides and angles. Shapes also belong to families inside families, like a family tree: every square is a rectangle, and every rectangle is a quadrilateral.
Key idea
A shape belongs to every category whose rules it meets. Categories nest: a smaller category (like squares) sits inside a bigger one (like rectangles) when every shape in it meets the bigger category's rules.
Why it works
Each category is defined by its properties. A rectangle is a quadrilateral with 4 right angles. A square has 4 right angles, so it meets that rule: it is a rectangle, a special one that also has 4 equal sides. Being special does not take a shape out of the bigger family.
| Shape | Its rule | Also always a |
|---|---|---|
| Parallelogram | 2 pairs of parallel sides | quadrilateral |
| Rectangle | 4 right angles | parallelogram |
| Rhombus | 4 equal sides | parallelogram |
| Square | 4 right angles and 4 equal sides | rectangle and rhombus |
Worked examples
Worked example: Most specific name
A quadrilateral has 4 equal sides but no right angles. What is its most specific name?
- 4 equal sides: it is a rhombus.
- No right angles, so it is not a square.
Answer: rhombus
Worked example: Where two families meet
Which shapes are both rectangles and rhombuses?
- A rectangle has 4 right angles; a rhombus has 4 equal sides.
- A shape with both is a square.
Answer: squares
Common mistake
Thinking a shape can only have one name. A square is also a rectangle, a rhombus, a parallelogram and a quadrilateral. Check by testing the rules: does it have what that family requires?
Some books define a trapezoid as having exactly one pair of parallel sides, and others as having at least one pair. With the second definition, every parallelogram is also a trapezoid. Either way, the family-tree idea is the same.