The quadrilateral family

The idea

Rectangles and rhombuses are parallelograms, and a square is a rectangle, a rhombus and a parallelogram; in a Venn diagram, the set of squares lies in the intersection of the sets of rectangles and rhombuses.

Many names for one shape

We now know four kinds of quadrilateral: parallelogram, rectangle, rhombus and square. One shape can have more than one name.

How many of the names parallelogram, rectangle, rhombus fit a square?

• Just one of them
• Two of them
• All three

Make all four kinds

Drag the corners B and D. The dot in the Venn picture shows which names fit your shape. Where does the dot go when the angle is 90°? When the sides are equal? When both?

Squares sit in the overlap

Rectangles and rhombuses are parallelograms, and a square is a rectangle, a rhombus and a parallelogram: in a Venn diagram, the set of squares lies in the intersection of the sets of rectangles and rhombuses.

The dot never left the big box: every shape was a parallelogram.

It entered the Rectangles circle at 90°, and the Rhombuses circle when the sides were equal. For both at once, it sat in the overlap: the squares.

So a square has the properties of all three: rectangle, rhombus and parallelogram.

Notes

Rectangles and rhombuses are parallelograms, and a square is a rectangle, a rhombus and a parallelogram: in a Venn diagram, the squares lie in the intersection of the rectangles and the rhombuses.

Check yourself

Which statement is true?

Where do the squares lie in the Venn diagram of rectangles and rhombuses?

A quadrilateral has four right angles and four equal sides. Which names fit it?

A quadrilateral has all sides equal and one angle of 60°. Which name fits it?

• Every rectangle is a rhombus. A rectangle that is 6 cm by 4 cm has four right angles, but its sides are not all equal. It is not a rhombus.
• Every rhombus is a parallelogram — correct. Yes! A rhombus has parallel opposite sides, so it is a parallelogram.
• Every parallelogram is a rectangle. A leaning parallelogram does not have right angles, so it is not a rectangle.
• In the part where the two circles overlap — correct. Yes! A square is both a rectangle and a rhombus, so it lies in the intersection of the two sets.
• Only in the Rectangles circle, outside the Rhombuses circle. A square has four equal sides, so it is a rhombus too. It lies in both circles.
• Outside both circles. A square is both a rectangle and a rhombus, so it lies inside both circles.
• Square, rectangle, rhombus and parallelogram — correct. Yes! Right angles and equal sides make it a square, and a square is also a rectangle, a rhombus and a parallelogram.
• Only square. It is a square, but every square is also a rectangle, a rhombus and a parallelogram.
• Rectangle and rhombus, but not a parallelogram. Rectangles and rhombuses are parallelograms, so a square is one too.
• A rhombus, which is also a parallelogram, but not a rectangle — correct. Yes! Equal sides make it a rhombus, so it is a parallelogram. Its angle is 60°, not 90°, so it is not a rectangle.
• A square. A square needs all its angles to be 90°. This one has a 60° angle.
• A rectangle. A rectangle needs four right angles. This one has a 60° angle.