Properties of 2D shapes
Triangles
| Triangle | Sides | Angles | Lines of symmetry |
|---|---|---|---|
| all 3 equal | all 60° | 3 | |
| 2 equal | 2 equal | 1 | |
| all different | all different | 0 | |
| Right-angled | any | one is 90° | 0 or 1 |
A right-angled triangle can also be isosceles: its angles are 90°, 45° and 45°.
Example: a triangle has angles 30°, 75° and 75°. What type of triangle is it?
- Two angles are equal, so two sides are equal.
- Isosceles
Key terms
Quadrilaterals: sides and angles
| Shape | Sides | Angles |
|---|---|---|
| Square | 4 equal; opposite sides parallel | all 90° |
| Rectangle | opposite sides equal and parallel | all 90° |
| Rhombus | 4 equal; opposite sides parallel | opposite angles equal |
| Parallelogram | opposite sides equal and parallel | opposite angles equal |
| Kite | two pairs of equal sides next to each other | one pair of opposite angles equal |
| Trapezium | one pair of parallel sides | — |
Example: a quadrilateral has 4 equal sides but no right angles. What is it?
- 4 equal sides: a square or a rhombus.
- No right angles, so not a square.
Quadrilaterals: diagonals and symmetry
| Shape | Diagonals | Lines of symmetry | Rotational symmetry |
|---|---|---|---|
| Square | equal; cross at 90°; bisect each other | 4 | order 4 |
| Rectangle | equal; bisect each other | 2 | order 2 |
| Rhombus | cross at 90°; bisect each other | 2 | order 2 |
| Parallelogram | bisect each other | 0 | order 2 |
| Kite | cross at 90° | 1 | order 1 |
| Isosceles trapezium | equal | 1 | order 1 |
"Bisect each other" means each diagonal cuts the other in half.
Example: which quadrilaterals have diagonals that cross at right angles?
Symmetry
- A shape has if a mirror line splits it into two halves that are reflections of each other.
- The is how many times the shape looks the same during one full turn. A shape that only fits once has order 1 (no rotational symmetry).
A regular polygon with n sides has n lines of symmetry and rotational symmetry of order n.
Example: give the number of lines of symmetry and the order of rotational symmetry of a regular hexagon and of a parallelogram.
- Regular hexagon: 6 lines, order 6
- Parallelogram: 0 lines, order 2 (it fits again after a half turn)
Common mistake
A parallelogram's diagonals are not lines of symmetry: folding along one doesn't make the halves match.