Math Detours · Grade 5 · Sharpen · 15

Is a square a rectangle?

Yes, and here is why. Shapes belong to families, and a family is defined by a rule, not by a look.

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Tonight's idea

Yes, a square is a rectangle. Here is why.

In math, a shape gets its name from a rule, not from how it looks. The rule for a rectangle is short: four straight sides and four right angles. A right angle is a square corner, like the corner of this page.

The rule does not say "two long sides and two short sides". So test a square. Four straight sides? Yes. Four right angles? Yes. It passes, so it is a rectangle. It is a special one, because its sides are also all equal.

Quadrilaterals: 4 straight sides Parallelograms: 2 pairs of parallel sides Rectangles 4 right angles Rhombuses 4 equal sides Squares
Each loop is a family with its rule. A shape inside a loop is also inside every loop around it. Squares obey both rules, so they sit in the overlap.

It works like dogs and animals. Every dog is an animal, but not every animal is a dog. Every square is a rectangle, but not every rectangle is a square.

Two more words. Parallel sides run side by side and never meet, like train tracks. A rhombus has four equal sides. It can lean over.

Watch one: always, sometimes or never?

"A square is a rhombus." The rule for a rhombus is four equal sides. Every square has four equal sides. So this is always true.

"A rhombus is a square." A square needs four right angles too. A rhombus that leans over has none, but an upright one has four. So this is sometimes true.

To decide, do not look at the picture in your head. Read the rule and test it.

Your turn

  1. Here are five names: quadrilateral, parallelogram, rectangle, rhombus, square. Which names fit each shape? (a) Four right angles, with sides of 3 cm, 5 cm, 3 cm, 5 cm. (b) Four right angles, with every side 4 cm.
  2. Always, sometimes or never? (a) A square is a parallelogram. (b) A rectangle has four equal sides. (c) A rhombus has exactly one right angle. (d) A triangle is a quadrilateral.
  3. Draw each one. (a) A rhombus that is not a square. (b) A parallelogram that is not a rectangle and not a rhombus. (c) A quadrilateral that is not a parallelogram.
  4. Here is a new family. A kite has four sides in two pairs. The two sides in each pair are equal and they touch each other. (a) Is a rhombus a kite? (b) Is a rectangle with sides 3 cm, 5 cm, 3 cm, 5 cm a kite? Give your reasons.

Stretch: nine dots

Draw nine dots in a 3 by 3 grid. How many squares can you draw with a dot at every corner? The squares can be different sizes. One of them is tilted. Remember: it is a square by its rule, not by its look.

Answers

  1. (a) Quadrilateral, parallelogram and rectangle. Not rhombus or square, because the sides are not all equal. (b) All five names fit.
  2. (a) Always. A square is a rectangle, and every rectangle is a parallelogram. (b) Sometimes: only when it is a square. (c) Never. If a rhombus has one right angle, all four are right angles, and it is a square. (d) Never. A triangle has three sides.
  3. (a) Any leaning shape with four equal sides, like a diamond. (b) A leaning shape with two pairs of parallel sides, one pair longer than the other. (c) Any four-sided shape with a pair of opposite sides that are not parallel.
  4. (a) Yes. All four sides are equal, so any two that touch are equal. Both pairs pass the rule. (b) No. Sides that touch are 3 cm and 5 cm, so they are never equal. The equal sides are opposite each other.

Stretch. Six. There are four small squares and one big square round the outside. The sixth joins the middle dots of the four edges. It stands on a corner like a diamond, but it has four equal sides and four right angles.

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