Quadrilaterals

A family tree of shapes with four corners

You have known a square and a rectangle since you were little. Today you meet the whole family they belong to — quadrilaterals — and you learn to read the family relationships.

A quadrilateral is any shape with four sides and four vertices. The sum of its interior angles is 360° (twice 180°, because a diagonal splits it into two triangles).

The family branches by parallel sides. When both pairs of opposite sides are parallel, it is a parallelogram. When only one pair, it is a trapezium. When none, it is a general quadrilateral.

And parallelograms have their subfamilies: a rectangle (right angles), a rhombus (equal sides), and a square, which is both at once — right angles and equal sides. That is why there are sentences that sound like riddles: every square is a rectangle, but not every rectangle is a square. Today we sort that out once and for all.

Filip and Sofie sort cut-out shapes for a display. Filip holds a rhombus: “That is a squashed square, right?” Sofie: “Almost. It has four equal sides like a square, but the angles need not be right.” She tries it the other way: “And a square is a rhombus whose angles lined up at ninety.” She draws a family tree on paper: at the top a quadrilateral, under it a trapezium and a parallelogram, under the parallelogram a rectangle and a rhombus, and right at the bottom a square with arrows from both. “A square is a noble,” Filip laughs, “it inherited everything from two sides of the family.”

Who is who among quadrilaterals

Parallelogram — both pairs of opposite sides parallel. From that it follows: opposite sides are equally long, opposite angles equal, the diagonals bisect each other. Picture a collapsed rectangle — a frame of battens without the bar from lesson 18.

Rectangle — a parallelogram with right angles extra. The diagonals are equally long.

Rhombus — a parallelogram with all sides equal. The diagonals are perpendicular and bisect its angles.

Square — a rectangle and a rhombus at once: right angles and equal sides. It inherited all the properties of both.

Trapezium — only one pair of parallel sides; they are called the bases, the other two the legs. An isosceles trapezium has equally long legs (the shape of a bridge pier or a lampshade).

General quadrilateral — no parallel sides. For example a kite for flying (a deltoid) with two pairs of neighbouring equal sides.

An angle check: in every quadrilateral the four interior angles add up to 360°. You know three angles → you calculate the fourth.

Read the family tree from top to bottom as “is a special case of”: a square is a special rectangle, a rectangle a special parallelogram, a parallelogram a special quadrilateral.

Calculate the fourth angle of a quadrilateral: 85°, 110° and 95°

Step 1: the sum of the angles of a quadrilateral is 360°.

Step 2: I add the known ones: 85 + 110 + 95 = 290°.

Step 3: the fourth angle: 360 − 290 = 70°.

Result: 70°. Check: 85 + 110 + 95 + 70 = 360. That checks out. Why 360? A diagonal splits a quadrilateral into two triangles of 180° each — and 2 × 180 = 360. Geometry rules stack like Lego from the ones you already know.

True or false? Three claims about the family tree

a) “Every square is a rectangle.” True — a square has right angles and opposite sides parallel, it satisfies everything a rectangle asks.

b) “Every rectangle is a square.” False — a rectangle is missing the equal-sides condition. One counterexample is enough: a rectangle 3 × 5.

c) “A rhombus is a parallelogram.” True — it has both pairs of sides parallel.

The lesson: an arrow in the family tree works only one way. Down (from general to special) properties are added, not taken away.

Spot the shape by its diagonals

A detective question: a quadrilateral has diagonals that bisect each other, are equally long, but are not perpendicular. Who is it?

They bisect each other → it is a parallelogram. Equally long → a rectangle. Not perpendicular → it is not a square (there they would have to be, a square is also a rhombus).

Result: a rectangle that is not a square. Diagonals are a quadrilateral’s fingerprint — each kind has them different and by them you can spot it even without the sides.

Practice: now you do it 🎲

Now try it for real. You will see all the questions at once, like on paper. 시작 with ten. When it goes well, add more.

Revise the area of a rectangle — the next lesson builds on it. Write only the number in cm².

Practise the questions

On paper: a family tree and detective work

Draw freehand, but write exact properties. This is about sorting, not drawing.

1. Draw a family tree of quadrilaterals: quadrilateral → trapezium / parallelogram → rectangle / rhombus → square. By each write one spotting property.

2. Calculate the fourth angle: a) 90°, 90°, 120°; b) 75°, 105°, 88°. At a) decide whether it can be a parallelogram.

3. Decide true/false and explain: “Every rhombus is a square.” — “A square is a trapezium.” — “A parallelogram has opposite angles equal.”

4. Draw a quadrilateral whose diagonals are perpendicular and bisect each other, but are not equally long. What is it called?

Now you 💪

1. I sort quadrilaterals by pairs of parallel sides.
2. I know the special features of a rectangle, a rhombus and a square and their inheritance.
3. I calculate the fourth angle from 360°.

What to take from this lesson

• The sum of the angles of a quadrilateral is 360°.
• Parallelogram: two pairs of parallel sides. Trapezium: one pair.
• Rectangle = parallelogram with right angles, rhombus = with equal sides.
• A square inherits the properties of a rectangle and a rhombus.
• The family tree holds one way: every square is a rectangle, the other way not.

© 2026 Ing. Martin Polak / AlgoRhino · 콘텐츠 이용 약관