Square

The idea

A square is a quadrilateral whose angles are all 90° and whose sides are all equal; every square is a rectangle, but not every rectangle is a square, so all rectangle properties hold for a square.

Is a square a rectangle?

A rectangle has four right angles and equal opposite sides. A square has four right angles and all four sides equal.

Is a square also a rectangle?

• Yes, every square is a rectangle
• No, a square is a different shape
• Only some squares are

Stretch a rectangle

Drag the corner C. Watch the dot in the picture, and the two chips: are the angles still 90°? When are the sides all equal?

Squares sit inside rectangles

A square is a quadrilateral whose angles are all 90° and whose sides are all equal. Every square is a rectangle, but not every rectangle is a square, so all the properties of a rectangle hold for a square.

The angles stayed 90° all the time. Only the sides changed: when they became equal, the dot moved into the Squares circle, which lies inside the Rectangles circle.

So a square passes the rectangle test, and every fact about rectangles is also a fact about squares: equal diagonals, diagonals that bisect each other, opposite sides that are parallel.

Notes

A square is a quadrilateral whose angles are all 90° and whose sides are all equal. Every square is a rectangle, but not every rectangle is a square.

Check yourself

Which statement is true?

A quadrilateral has four right angles and sides 5 cm, 5 cm, 5 cm, 5 cm. What can we call it?

ABCD is a square. The diagonal AC = 10 cm. How long is the diagonal BD?

Answer: 10 cm

A square is a rectangle, so the properties of a rectangle hold: its diagonals are equal. BD = AC = 10 cm.

A rectangle has a perimeter of 36 cm, and its neighbouring sides are equal. How long is each side?

Answer: 9 cm

Equal neighbouring sides make all four sides equal: it is a square. Each side is 36 ÷ 4 = 9 cm.

• Every rectangle is a square. A rectangle that is 8 cm by 5 cm has four right angles, but its sides are not all equal. It is not a square.
• Every square is a rectangle — correct. Yes! A square has four right angles and equal opposite sides, so it passes the rectangle test.
• No square is a rectangle. A square has four right angles and equal opposite sides, so it is a rectangle.
• A square only. It is a square, but a square is also a rectangle, so that is not all we can say.
• A rectangle only. It is a rectangle, but its four sides are equal as well, so it is also a square.
• Both a square and a rectangle — correct. Yes! Four right angles and all sides equal: it is a square, and every square is a rectangle.