Any shape with four sides is a quadrilateral. They all share one fact — their angles add to 360° — and then each special one adds its own rules.
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Split any polygon into triangles from one corner. Each triangle gives 180°, and that is where the formula comes from.
📐 The two formulas
Interior angles of an n-sided polygon add to (n − 2) × 180°. For a quadrilateral: (4−2)×180 = 360°.
Exterior angles of any polygon always add to 360° — no matter how many sides.
For a regular polygon, each interior angle = (n−2)×180 ÷ n, and each exterior angle = 360 ÷ n.
📐 Worked out
| Polygon | Sides | Interior sum | Each angle if regular |
|---|---|---|---|
| Triangle | 3 | 180° | 60° |
| Quadrilateral | 4 | 360° | 90° |
| Pentagon | 5 | 540° | 108° |
| Hexagon | 6 | 720° | 120° |
| Octagon | 8 | 1080° | 135° |
| Decagon | 10 | 1440° | 144° |
Each one adds a rule to the one before it. A square is the end of the chain — it has every property.
⬜ Properties at a glance
| Shape | Sides | Angles | Diagonals |
|---|---|---|---|
| Trapezium | one pair parallel | — | — |
| Parallelogram | opposite sides equal and parallel | opposite angles equal | bisect each other |
| Rhombus | all four equal | opposite equal | bisect at 90° |
| Rectangle | opposite equal | all 90° | equal, bisect each other |
| Square | all four equal | all 90° | equal, bisect at 90° |
| Kite | two pairs of adjacent equal | one pair equal | one bisects the other at 90° |
The chain: every square is a rhombus and a rectangle; every rhombus and rectangle is a parallelogram; every parallelogram is a trapezium. It never runs the other way — a rectangle is not always a square.
Two favourite exam facts. In a parallelogram, adjacent angles are supplementary (they add to 180°). In a rhombus the diagonals are not equal, but they do cross at right angles.
Almost every question here is the same: use 360°, or use a property, then subtract.
Method: if three angles of a quadrilateral are 80°, 95° and 100°, the fourth is 360 − (80+95+100) = 85°. If it is a parallelogram and one angle is 70°, the next one is 180 − 70 = 110°, and the opposite one is also 70°.
12 questions on quadrilaterals. Draw the figure before you answer. 🍀
