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The quadrilateral family tree

Geometry · level 3, lesson 5 of 6

A question to start

Here is a riddle that starts arguments in classrooms. Is a square a rectangle?

Some people say "No! A rectangle is long and thin, and a square is not." Others say "Yes! It has four right angles, so it must be a rectangle."

Who is right? To decide, we have to say exactly what each word means. Once we do, the four-sided shapes fall into a family tree, with grandparents, parents and children, and the riddle answers itself.

The idea

Quadrilaterals

A quadrilateral is a flat shape with four straight sides ("quadri" means four, as in quad bike; "lateral" means side). It also has four corners (vertices) and four angles.

A quadrilateral can be any shape at all, as long as it has four straight sides: lopsided, squashed, even with one corner poking inwards like an arrowhead. Some quadrilaterals have special features, and those get special names.

The special features to look for are:

  • parallel sides: sides that point the same way and never meet (lesson 3);
  • equal sides: shown with tick marks;
  • right angles: shown with a small square.
square rectangle parallelogram rhombus trapezoid kite

The special quadrilaterals

Trapezoid: at least one pair of parallel sides. The two parallel sides are often called the bases. (In Britain this shape is called a trapezium. Some books say a trapezoid has exactly one pair of parallel sides; on this site we use "at least one pair", which keeps the family tree simple.)

Parallelogram: two pairs of parallel sides. Each side is parallel to the side opposite it. The word just means "parallel lines".

Rectangle: four right angles. ("Rect" comes from a word for upright or right, as in right angle.)

Rhombus (say "ROM-bus"): four equal sides. A rhombus often looks like a square that has been pushed over, or like a diamond on a playing card.

Square: four equal sides and four right angles.

Kite: two pairs of equal sides, where the equal sides are next to each other (not opposite). It looks like a kite flying in the sky: two short sides at the top, two long sides at the bottom.

Climbing the family tree

Now the important step. Read each definition as a list of things a shape must have, and ask: does one shape have everything on another shape's list?

  • A square has four right angles. That is everything on the rectangle's list. So every square is a rectangle. A very special one, with all sides equal too.
  • A square also has four equal sides. That is everything on the rhombus's list. So every square is a rhombus.
  • A rectangle has opposite sides parallel (both pairs are perpendicular to the same sides, like the rails of a ladder). So every rectangle is a parallelogram.
  • A rhombus also turns out to have opposite sides parallel. So every rhombus is a parallelogram.
  • A parallelogram has at least one pair of parallel sides, so every parallelogram is a trapezoid.
  • A rhombus has two pairs of equal sides next to each other (any two sides that meet are equal), so every rhombus is a kite.

Put that together and you get a family tree. Each shape is a special case of every shape above it that a line leads up to.

quadrilateral trapezoid kite parallelogram rectangle rhombus square

So the riddle at the start has a clear answer: yes, a square is a rectangle. It is a rectangle that happens to have all four sides equal. In the same way, you are a person, and you are also a child, and you are also a reader: one thing can belong to several families at once. Going down the tree, shapes become more special and have more features. Going up, they become more general and have fewer rules to follow.

The question "Is a rectangle a square?" has a different answer: not always. A rectangle 2 cm by 5 cm has four right angles but its sides are not all equal, so it is not a square. A shape only belongs to a family if it has everything on that family's list.

Extra features that come free

Definitions are kept short on purpose. But once a shape fits a definition, other features come along with it, free. Here are the most useful ones.

Parallelogram. Opposite sides are equal in length, and opposite angles are equal. You can see why by cutting a paper parallelogram along a diagonal (a line joining opposite corners): the two triangles fit exactly on top of each other after a half turn. Every side and angle of one triangle lands on its partner.

Rectangle. The two diagonals are equal in length. Flip a rectangle over like turning a page, left to right: it lands on itself, and one diagonal lands exactly on the other. So they are the same length.

Rhombus. The diagonals cross at a right angle, and each cuts the other exactly in half. A rhombus also folds in half along each diagonal.

Kite. The diagonals cross at a right angle. The diagonal that runs between the two corners where the equal sides meet is a line of symmetry: fold along it and the kite matches itself.

Square. Everything above. Its diagonals are equal, they cross at a right angle, and they cut each other in half.

rectangle: equal diagonals rhombus: cross at 90°, cut each other in half kite: cross at 90°

The angles add to 360°

Draw any quadrilateral and join two opposite corners with a diagonal. It cuts the shape into two triangles. The angles of the two triangles together make up exactly the four angles of the quadrilateral. Each triangle's angles add to 180° (the fact you will prove in level 4), so

the four angles of any quadrilateral add up to \(180 + 180 = 360\) degrees.

Check it on a rectangle: \(90 + 90 + 90 + 90 = 360\). ✓

Worked examples

1. Guess my shape. "I have four equal sides, but my angles are not right angles." Four equal sides makes it a rhombus. Not having right angles means it is not a square. Answer: a rhombus (and so also a parallelogram, a trapezoid and a kite, but "rhombus" is the most exact name).

2. Always, sometimes or never?

  • "A rectangle is a parallelogram." Always: every rectangle has two pairs of parallel sides.
  • "A parallelogram is a rectangle." Sometimes: only when its angles are right angles.
  • "A kite is a parallelogram." Sometimes: a rhombus is both. A tall kite with short top sides and long bottom sides is not, because its opposite sides are not parallel.
  • "A trapezoid has four right angles." Sometimes: a rectangle is a trapezoid with four right angles, but most trapezoids do not.

3. A missing angle in a parallelogram. One angle of a parallelogram is 70°. Find the other three.

  1. Opposite angles are equal, so the angle across from it is also 70°.
  2. The other two angles are equal to each other. Call each one "?".
  3. All four add to 360: \(70 + 70 + ? + ? = 360\), so \(? + ? = 360 - 140 = 220\).
  4. Each is \(220 \div 2 = 110\) degrees.

The angles are 70°, 110°, 70°, 110°. Notice that two angles next to each other add to \(70 + 110 = 180\) degrees. That happens in every parallelogram, and in level 4 you will see why using parallel lines.

4. The most exact name. A shape has two pairs of parallel sides and four right angles, and its sides are 3 cm, 6 cm, 3 cm, 6 cm. It is a parallelogram (two pairs of parallel sides) with four right angles: a rectangle. The sides are not all equal, so it is not a square.

Try it

  1. What is the most exact name for a quadrilateral with: (a) four equal sides and four right angles; (b) exactly one pair of parallel sides; (c) two pairs of parallel sides and no right angles, with sides 4, 7, 4, 7 cm?
  2. Always, sometimes or never? (a) A square is a rhombus. (b) A rhombus is a square. (c) A trapezoid is a parallelogram. (d) A rectangle has two diagonals of different lengths.
  3. Three angles of a quadrilateral are 90°, 90° and 120°. What is the fourth?
  4. One angle of a rhombus is 50°. Find the other three. (A rhombus is a parallelogram.)
  5. Draw the family tree from memory. Which shape sits at the very bottom, and why?
  6. Mia says: "My shape is a rectangle and a rhombus." What shape must it be?
  7. A kite has one angle of 120° between its two short sides and one angle of 60° between its two long sides. Its other two angles are equal. What are they?
Answers
  1. (a) Square. (b) Trapezoid (only one pair parallel, so not a parallelogram). (c) Parallelogram: the sides are not all equal (so not a rhombus) and there are no right angles (so not a rectangle).
  2. (a) Always: a square has four equal sides. (b) Sometimes: only when its angles are right angles. (c) Sometimes: only when both pairs of sides are parallel. (d) Never: a rectangle's diagonals are always equal.
  3. \(360 - 90 - 90 - 120 = 60\) degrees.
  4. The opposite angle is also 50°. The other two share \(360 - 100 = 260\), so each is 130°. The angles are 50°, 130°, 50°, 130°.
  5. The square is at the bottom. It has every feature: four right angles (rectangle), four equal sides (rhombus), parallel opposite sides (parallelogram), at least one pair of parallel sides (trapezoid) and equal sides next to each other (kite). So it belongs to every family above it.
  6. A square. Being a rectangle gives it four right angles; being a rhombus gives it four equal sides. Both together is exactly a square.
  7. The four angles add to 360: \(120 + 60 = 180\), so the other two share \(360 - 180 = 180\) degrees, and each is 90°.

Common mistake

Thinking each shape has only one name. A square is also a rectangle, a rhombus, a parallelogram, a trapezoid and a kite. When asked for the most exact name, give the lowest one on the tree. When asked "is it a ...?", check the definition's list, not what the shape usually looks like.

Thinking a rhombus must stand on its point. A diamond standing on its corner and the same rhombus turned to lie on a side are the same shape. A square turned 45° is still a square, even though people sometimes call it a diamond. Turning a shape never changes its name.

Where it leads